Ordinal Compactness

ORDINAL COMP A CTNESS P A OLO LIPP ARINI Abstract. W e int ro duce a new covering property , deﬁned in terms of order t ypes of sequences of op e n sets, rather than in terms of cardinalities o f families. The most gener al form of this compa c t- ness notion depe nds on tw o or dinal parameters . In the particular case when the para meters are c a rdinal num ber s, we get bac k a classical notio n. Generalized to ordinal num bers, this notion turns out to b ehav e in a m uch more v ar ied wa y . W e prov e many nontrivial results o f the form “e very r α, β s -compact space is r α 1 , β 1 s -compact”, for ordinals α , β , α 1 and β 1 , while only trivia l results o f the ab o ve form hold, if we re s trict to cardinals. Counterexamples are pr o vided showing that our results a r e optimal. W e present many e xamples of spa ces s atisfying the very same cardinal compac tnes s proper ties, but with a bro ad ra nge o f distinct behaviors, with r espect to ordinal compactness. A muc h more r e - ﬁned theor y is obtained for T 1 spaces, in compariso n with arbitrar y top ological spa c e s. The notio n of o rdinal compactness becomes partly trivial for spaces of small car dinalit y . 1. Introduction The no w ada ys standard notio n of c omp actness for top ological spaces is usually expressed in terms of cardinalities of op en cov ers, and as- serts t ha t every op en cov er has a ﬁnite sub co v er. Since compact spaces constitute a relativ ely special class, v a rious w eak enings hav e been ex- tensiv ely considere d, the most notable b eing Lin del¨ ofness (“an y op en co v er has a coun table sub co v er”), and c ountable c omp actness (“an y coun table op en co v er ha s a ﬁnite sub cov er”). Still more g enerally , ﬁna l κ -c om p actness asserts that any op en cov er has a sub co v er of cardinality  κ , and initial κ -c omp actness asserts that ev ery op en cov er of cardi- nalit y ¤ κ has a ﬁnite sub co v er. A v ast literature exists on the sub ject: see the surv eys [Go, Ste, V3, V4], and, as a v ery sub jectiv e and partial 2000 Mathematics Subje ct Classiﬁc ation. P r imary 5 4D20, 03E 1 0; Secondary 54A05, 54D10. Key wor ds and phr ases. Ordina l, co mpa ctness, o pen cover, top ological space, T 1 , disjoint union, shifted sum. 1 2 P AOLO LI PP AR INI c hoice, [BN, ScT a, S hTs, T] for more r ecen t line s of r esearch . See also the references there. In this note w e extend the notio n of cardinal compactness to ordinals, that is, w e take into accoun t order types of fa milies o f cov erings, rather than just their cardinalities. Assuming the Axiom of Choice, eac h cardinal can b e seen as an ordinal, th us o ur notion is more general: when a sequence is cardinal-lik e ordered, we get bac k the more usual notions. On the con trary , and quite surprisingly , it turns out that our ordinal g eneralization pro vides a muc h ﬁner tuning of compactness prop erties of top ological spaces. 1.1. A ﬁrst example: Lindel¨ of nu m b ers. Before discussing the most general ve rsion of our notion, let us exemplify it in the particular case of Lindel¨ of n um b ers. Let us deﬁne the Lindel¨ of  c ar dinal of a top ological space X as t he smallest c ar dinal λ suc h tha t ev ery op en co v er of X has a sub co v er of cardinality  λ (the superscript  is a reminder t hat the more common deﬁnition asks just for a sub co v er of cardinalit y ¤ λ . T he presen t v arian t is more conv enien t here, since it distinguishes b et w een compactness and Lindel¨ ofness). In other words , the Lindel¨ of  cardinal of a top ological space is the smallest cardina l λ suc h that the space is ﬁnally λ -compact. As an ordinal generalization o f the ab o v e notion, let us deﬁne the Lindel¨ of or dinal of a t o po lo gical space X as the smallest or dinal α suc h that, for ev ery o pen co v er of X whose elemen ts are indexed b y some ordinal β , there exists some subset H of β suc h that H has order t yp e  α , and the set of elemen ts with index in H still constitutes a cov er of X . Thus w e are dealing with cov ers ta k en in a certain (w ell) order and, when dealing with sub co v ers , w e w an t the order of the original co v er to b e resp ected. While the Lindel¨ of o r dina l o f a space clearly determines its Lindel¨ of  cardinal, on the con trary , there are spaces with the same Lindel¨ of  car- dinal, but with v ery diﬀeren t Lindel¨ of ordinals. As a simple example, if κ is a regular uncoun table cardinal, then κ , b oth with the discrete top ology , and with the order top olog y , has Lindel¨ of  cardinal κ  . On the other hand, though κ  is also the Lindel¨ of or dina l o f the former space, the latter space has a muc h smaller Lindel¨ o f ordinal, that is, κ  ω (here and b elo w,  denotes or dinal sum ). In termediate cases can o ccur: for example, the disjoin t union of tw o copies of κ with the order top ology has Lindel¨ of o rdinal κ  κ  ω . W e can also hav e κ  1, κ  2, . . . as Lindel¨ of o rdinals, but o nly in some pathological cases, and only for spaces satisfying v ery few separation prop erties. More in v olv ed ex- amples shall b e presen ted in the b o dy of the pap er. Th us our ordinal ORDINA L COMP ACT NESS 3 generalization can b e used to distinguish among spaces whic h app ear to b e quite similar, as f ar as the cardinal notio n is considered. Imp osing further conditions on a space pro vides some constrain ts on its Lindel¨ of ordinal. F or example, the Lindel¨ of o rdinal of a coun table space is either ω 1 , or is ¤ ω  ω . F or spaces of cardinality κ , there are similar limitations, sligh tly more in v olv ed. Stronger restrictions are obtained by imp osing mild separation axioms. F or example, the Lindel¨ of ordinal of a T 1 space (of any car dinality) is either ¤ ω , or ¥ ω 1 . Ac tually , only ordinals o f a v ery sp ecial form can b oth ha v e coﬁnalit y ω and b e the Lindel¨ of ordinal of some T 1 space (Corollary 6.11). W e also sho w that, for arbitrary spaces, the Lindel¨ of ordinal of a disjoint union is exactly determined b y the Lindel¨ of ordinals of the summands. Summing up, the Lindel¨ of ordina l of a top olog ical space app ears to b e a quite ﬁne measure of the compactness prop erties the space satisﬁes. Moreov er, there are interesting and deep connections b etw een the p ossible v alues the Lindel¨ of ordinal can take , and cardinalities and separation prop erties of spaces. 1.2. r µ, λ s -compactness (for cardinals). No w w e pro ceed b y con- sidering more general forms of compactness . All the (cardinal) com- pactness prop erties deﬁned in the ﬁrst paragra ph of this introduction can b e uniﬁed in a single framew ork b y in tro ducing the f ollo wing tw o- cardinals prop ert y . F or cardinals µ ¤ λ , a to p olo gical space is said to b e r µ, λ s -c om p act if and only if ev ery op en co v er by at most λ sets has a sub co v er with  µ sets. Th us, for example, compactnes s is the same as r ω , λ s -compactness, for ev ery cardinal λ , and Lindel¨ ofness is r ω 1 , λ s -compactness, for ev ery cardinal λ . On the other hand, count- able compactness is r ω , ω s -compactness, and, more generally , initial λ -compactness is r ω , λ s -compactness. With a restriction on regular cardinals, and also in v arious equiv alent forms, the ab ov e t w o-cardinals ve rsion has b een in tro duced in 1929 b y P . Alexandroﬀ and P . Urysohn [A U]. F or arbitrary cardinals, the v ery exact form of the ab o v e deﬁnition seems to hav e ﬁrst app eared in [Sm]. It has b een studied b y man y p eople, sometimes under diﬀerent names and notations, a nd in sev eral equiv alent formulations. See a surv ey o f further related notions and results in [V2]. See also, e. g., [Ga, Li4, V1] and references there for f ur t her information. Apart from intrinsic in terest, r µ, λ s -compactness has pro v ed useful in man y cases. Beside s prov iding a common generalization of coun t- able compactness, Lindel¨ ofness, a nd so on, it exhibits a v ery in teresting 4 P AOLO LI PP AR INI feature: r µ, λ s -compactness is equiv alent to r ν, ν s -compactness, fo r ev- ery ν with µ ¤ ν ¤ λ . In par t icular, (full) compactness is equiv alent to r ν, ν s -compactness for ev ery inﬁnite cardinal ν , and Lindel¨ ofness is equiv alen t to r ν, ν s -compactness for ev ery ν ¡ ω . In other words, we can “slice” compactness in to smaller pieces. This fact has found many applications, mainly in view of the fact that, for ν regular, r ν, ν s -com- pactness has man y equiv alent formulations, most notably in terms of the existence of accumu lation p oin ts of sets of cardinality ν . See [V1]. See a lso [Li2], where the ab ov e mentioned “slicing” pro cedure has found another substan tial application. By the wa y , let us also men tion that the notion of r µ, λ s -compact- ness has ostensibly inspired some f urther notions out side mainstream general top ology . Most notably , some of the earliest deﬁnitions of b oth w eakly and strongly compact cardinals w ere introduced as forms of r κ, κ s -compactness for certain inﬁnitary languages [J, Chapters 17 a nd 20]. The exact top ological conte n t of these deﬁnitions later clearly emerged: see [Cai2, Ma] for history , references, and for ot her notions in Mo del Theory and Lo g ic whic h hav e apparently b een inspired by r µ, λ s -compactness. Also the notion of a p µ, λ q -regular ultraﬁlter, whic h pla y ed some ro le in the ev olution of Set Theory [CN ], [KM, Section 13], [J, p. 373], appar en tly originat ed in this stream of ideas. 1.3. The ordinal generalization. Motiv ated b y the in terest of (car- dinal) r µ, λ s -compactness, we started considering the p ossibilit y of an ordinal generalization. Though initially misled by the observ atio n that “initial α -compactness” actually reduc es to a cardinal notion (Corol- lary 2.8), w e so on r ealized that the more general notion of “tw o or dina ls compactness” is really new, as exempliﬁed ab o v e in the particular case of Lindel¨ of-like prop erties or, put in other w ords, ﬁnal α -compactness. In detail, if β and α are ordinals, let us sa y that a space X is r β , α s - compact if a nd only if ev ery α -indexed op en cov er has a sub co v er in- dexed b y a set o f order ty p e  β (in the induced order). Ordinal compactness, in the ab ov e sense, turns o ut to ha v e some v ery particular feat ures. As in the case of cardinals, w e can sho w that, also for ordinals, r β , α s -compactness is equiv alen t to r γ , γ s -compactness, for ev ery or dinal γ with β ¤ γ ¤ α . How ev er, the similarities ess en tially stop here. Indeed, for µ  λ inﬁnite regular cardinals, r µ, µ s -compact- ness and r λ, λ s -compactness are indep enden t prop erties. On the other hand, for ordinals, w e ha v e man y results whic h tie to g ether r β , α s -com- pactness and r β 1 , α 1 s -compactness, fo r v arious β , α , β 1 and α 1 . Just to state some of the simplest relatio ns, we hav e that , for α and β inﬁnite ordinals, ORDINA L COMP ACT NESS 5 (1) If β ¤ α , then r β , α s -compactness implies r β , α  1 s -compact- ness. (2) r β  α, β  α s -compactness implies r β  α  α, β  α  α s - compactness. (3) r α, α s -compactness implies b oth r β  α, β  α s -compactness and r β  α, β  α s -compactness. Ho w ev er, no t “eve rything” is pro v able, ev en for ordinals hav ing the same cardinalit y . Indeed, still presen ting only some simple examples: (4) r α  1 , α  1 s -compactness do es not imply r α, α s -compactness, in general. (5) r κ  ω , κ  ω s -compactness do es not imply r κ, κ s -compactness, in general. (6) r κ  κ, κ  κ s -compactness do es not imply r κ  κ, κ  κ s -compact- ness, in general. Th us, ordinal compactness is a highly nontrivial notion, in compar- ison with cardina l compactness. Moreov er, the ordinal compactness prop erties of a top ological space are deeply aﬀected b oth by its car- dinalit y and its separation prop erties. F or example , for κ an inﬁnite regular cardinal, any counte rexample to Clause (6) a bov e m ust b e of cardinalit y ¡ κ . On the other hand, no T 1 space can b e a coun terexam- ple to Clause (4). Conside ring the compactness prop erties of disjoin t unions inv o lv es some problems on ordinal arithmetic whic h ar e not completely trivial. T 1 spaces turn out to b e a somewhat neat dividing line: man y r ather o dd coun terexamples, p ossible in spaces lac king separation prop erties, cannot b e constructed using T 1 spaces. Thus w e prov ide a quite neat theory for T 1 spaces. In particular, in this resp ect, countable ordinals b eha v e v ery diﬀeren tly from uncoun table ones. The compactness the- ory for T 1 spaces is trivial on coun table ordinals; more generally , apart from a few exce ptions, the o r dinal prop erties of a T 1 space are “in- v ariant” mo dulo in terv a ls o f coun table length. Apparen tly , assuming stronger separation axioms do es not seem to mo dify the theory a lot ; at larg e, w e get essen tially the same results and coun terexamples for T 1 and f o r normal spaces. Ho w ev er, there is still r o o m for the p ossibilit y of some ﬁner results holding only fo r normal spaces; this is left as an op en pro blem. 1.4. Synopsis of the pap er. In summary , the pap er is divided as fo l- lo ws. In Section 2 w e in tro duce the main deﬁnition, together with some relativ ely simple pro perties and a couple of equiv alent reformulations. Then w e prov e many r esults of the form “ev ery r α, β s -compact space is 6 P AOLO LI PP AR INI r α 1 , β 1 s -compact”; most of these results shall b e used in the rest of the pap er. In Section 3 w e then pro vide a lot of examples, show ing that r β , α s -compactness, for α and β ordinals, provides a v ery ﬁne tuning of prop erties of op en co v erings: there are man y spaces whic h sho w a v ery diﬀeren tiated b eha vior with resp ect to ordinals, but b ehav e ex- actly the same wa y , when α and β are take n to v ary only on cardinals. W e also sho w tha t man y o f the results of Section 2 a re the b est p ossible ones. The most basic example s are presen ted in Subsection 3.1; t hen in Subsection 3.2 w e discuss the b eha vior of ordinal compactness with resp ect to disjoin t unions, a nd show that many more countere xamples can b e obtained in suc h a w a y . W e also in tro duce a generalized form of inﬁnite disjoin t union with a partial compactiﬁcation. Compactness prop erties of disjoin t unions are shown to b e connected to some no- tions in or dina l arithmetics related to natural sums of ordinals. Suc h matters are clariﬁed in detail in Subsection 3.3. In Section 4 w e sho w that man y more implications b et w een compact- ness pr o perties hold, f or spaces of small cardinalit y; put in another w a y , certain coun terexample s can b e constructed only b y means of space s of suﬃcien tly lar g e cardinalit y . Suc h coun terexamples are indeed pro- vided in Section 5, where w e giv e an exact c haracterization of those pairs of ordinals α and β suc h that r α, α s -compactness implies r β , β s - compactness. In Section 6 w e then get a more reﬁned theory , which holds f or T 1 spaces. F or suc h spaces, r β , α s -compactness b ecomes tr iv- ial fo r countably inﬁnite ordinals (Corollary 6.8 ) . More generally , with a few exceptions, ordina l compactness for T 1 spaces is inv aria nt mo dulo in terv als of coun table length. F ina lly , Section 7 con tains v arious quite disparate remarks and problems. In particular, it introduces further generalizations of o rdinal compactness, and also discusses the p ossibil- it y of a v ar ia n t in a mo del theoretical sense. The presen t note by no means exhausts all that can b e said abo ut r β , α s -compactness. F urthermore, as w e mentioned, the notion of r β , α s - compactness can b e also g eneralized to diﬀeren t con texts. 2. Main definition and basic pr oper ties In this section w e in tro duce o ur main notion, and state some simple prop erties. W e compare it with the more usual notion whic h deals only with cardinals; then we start pro ving results of the form “ev ery r β , α s -compact space is r β 1 , α 1 s -compact”, for appropria te or dinals. F o r cardinal compactness, only trivial results of the ab o v e kind hold. In the subsequen t sections w e shall presen t coun terexamples sho wing that our results cannot b e impro v ed. ORDINA L COMP ACT NESS 7 Throughout, let α , β and γ b e nonzero ordinals, and λ , µ b e nonzero cardinals. As custom, w e shall assume the Axiom of Choice, hence we can iden tify cardinals with initial ordinals. Deﬁnition 2.1. If X is a nonempty set (usually , but not necessarily , a to po lo gical space), and τ is a nonempt y family of subsets of X , we sa y that p X , τ q is r β , α s -c om p act if and only if the fo llo wing condition holds. Whenev er p O δ q δ P α is a sequence of mem b ers of τ suc h that  δ P α O δ  X , then there is H  α with order t ype  β and suc h that  δ P H O δ  X . If there is no danger of confusion, we shall simply sa y X in place of p X , τ q . As usual, a sequence p O δ q δ P α of mem bers of τ suc h that  δ P α O δ  X shall b e called a c ov er of X . A sub c over of p O δ q δ P α is a subseque nce whic h itself is a cov er. By r β , α q -c om p actness we mean r β , α 1 s -compactness for all α 1  α . The nota t ion is justiﬁed b y Prop osition 2.3(4) b elo w. Another notatio n for r β , α q -compactness is r β ,  α s -compactness. Finally , r β , 8q -c om p actness is r β , α s -compactness for all ordinals α ¥ β . When α and β are b oth cardinals, a nd X is a top ological space ( τ b eing alw a ys understo o d to b e the top ology on X ), w e g et bac k the classical cardinal compactness notion o f Alexandroﬀ, Urysohn a nd Smirno v [A U, Sm ]. This is b ecause, for λ a cardinal, ha ving order type  λ is the same as ha ving cardinality  λ . Notice that we allow rep etitions in p O δ q δ P α , that is, w e allo w t he p os- sibilit y that O δ  O δ 1 , fo r δ  δ 1 . An equiv alen t and sometimes useful deﬁnition in which (among other things) rep etitions are not a llo w ed is giv en by Lemma 2.9. W e hav e given the deﬁnition in the presen t fo r m since it app ears somewhat simpler. R emark 2.2 . In the deﬁnition o f r β , α s -compactness, the assumption that the sequence is indexed b y elemen ts in the ordinal α is only for con v enie nce. W e get an equiv alen t deﬁnition b y a sking that, for ev ery w ell ordered set J of order type α , if p O j q j P J is a cov er of X , then there is H  J suc h that the order type o f H (under the order induced by the order on J ) is  β , and suc h that p O j q j P H is a co v er of X . Of course, r β , α s -compactness is equiv alen t to the following condition (just take complemen ts!). Whenev er p C δ q δ P α is a sequence of comple- men ts of mem b ers of τ , and  δ P H C δ  H , for ev ery H  α with order t yp e  β , then  δ P α C δ  H . 8 P AOLO LI PP AR INI As we shall see b elo w in Remarks 3.4 and 3 .11, ordinal compactness is actually a new no tion, tha t is, it cannot b e deﬁned in terms of cardinal compactness. W e ﬁrst list some simple but useful prop erties of r β , α s -compactness. Prop osition 2.3. L et α and β b e non z er o or din als. (1) If β ¤ β 1 and α 1 ¤ α then r β , α s -c om p actness implies r β 1 , α 1 s - c om p actness. (2) r β , α s -c om p actness is e quivalent to r γ , γ s -c om p actness for every γ with β ¤ γ ¤ α . (3) If β ¤ β 1 ¤ α , then X is r β , α s -c om p act if and only if X is b oth r β , β 1 q -c om p act and r β 1 , α s -c om p act. (4) r β , α q -c om p actness is e quivalent to r γ , γ s -c om p actness fo r every γ with β ¤ γ  α . Pr o of. (1) is trivial. If α 1  α , add dumm y elemen ts at the top of the sequence, for example, by adding new o ccurrences of one elemen t already in the sequence. One implication in (2) is immediate from (1). The con v erse is obtained by transﬁnite induction. Supp ose that X is r γ , γ s -compact, for eve ry γ with β ¤ γ ¤ α . W e shall pro v e r β , γ s - compactness, fo r ev ery γ with β ¤ γ ¤ α , b y induction on γ . The induction ba sis γ  β is true by assumption. As for the induction step, let β  γ ¤ α , and assume that X is r β , γ 1 s -compact, fo r ev ery γ 1 with β ¤ γ 1  γ . Let p O δ q δ P γ b e a co v er of X . By r γ , γ s -compactness, p O δ q δ P γ has a sub co v er S whose index set has order ty p e γ 1  γ . If γ 1  β , w e are done. Otherwise, b y r β , γ 1 s -compactness, and R emark 2.2, w e get a sub co v er of S whose index set has order type  β , a nd the item is prov ed. (3) The only if condition is immediate from (1). F or the con v erse, notice t hat, again b y (1), r β , β 1 q -compactness implies r γ , γ s -compact- ness for ev ery γ with β ¤ γ  β 1 , and that r β 1 , α s -compactness implies r γ , γ s -compactness for ev ery γ with β 1 ¤ γ ¤ α . Th us w e get r γ , γ s -compactness, for ev ery γ with β ¤ γ ¤ α , hence r β , α s -compactness, by (2). (4) is immediate f r om (2).  R emark 2.4 . When α , β , α 1 . . . are restricted to v a r y only on cardinals, rather tha n ordinals, Prop osition 2.3 still holds, with the same pro of. In fact, f o r inﬁnite cardina ls, (1) and (2) are classical r esults ab out r µ, λ s - compactness. Again for inﬁnite cardina ls, it is w ell kno wn (and easy to prov e) that, for top o logical spaces, r cf µ, cf µ s -compactness implies ORDINA L COMP ACT NESS 9 r µ, µ s -compactness. An o rdinal generalization of the ab ov e f a ct will b e giv en in Corollary 2.6 (8). F or inﬁnite regular cardinals, there is no other non trivial implication b et w een r µ, λ s -compactness and r µ 1 , λ 1 s -compactness, except for those whic h follo w immediately from the ab ov e men tio ned facts. Indeed, if λ is a regular inﬁnite cardinal, then λ , with the order top ology , is not r λ, λ s -compact, but it is r µ, µ s -compact for ev ery inﬁnite cardinal µ  λ . Hence, if µ ¤ µ 1 are inﬁnite cardinals, then λ with the order top ology is r µ, µ 1 s -compact if and only if λ R r µ, µ 1 s . (Here and in what follows r µ, µ 1 s shall denote t he in terv al consisting of those ordinals δ suc h that µ ¤ δ ¤ µ 1 .) More generally , the exact ordinal compactness prop erties of λ (with v a rious top ologies) shall b e determined in Example 3.2. Con trary to the case of cardinal compactness, and quite surprisingly , there are man y non trivial “transfer prop erties” for ordinal compact- ness, relating r β , α s -compactness and r β 1 , α 1 s -compactness, for v arious β , α , β 1 and α 1 . The next prop osition and its coro llary list some simple relations. More signiﬁcan t results along this line, and some c haracter- izations shall b e prov ed in Section 5 . Prop osition 2.5. Supp ose that β , α , β 1 and α 1 ar e nonze r o or dinals, and that ther e e x ists an i n je ctive function f : α 1 Ñ α such that, fo r every K  α with or der typ e  β , it happ ens that f  1 p K q has or d er typ e  β 1 . Then r β , α s -c om p actness implies r β 1 , α 1 s -c om p actness. The assumption that f is inje c tive c an b e dr opp e d in the c ase of top olo gic al sp a c es (or just assumin g that τ is close d under unions). Pr o of. Supp ose that p X , τ q is r β , α s -compact, and let f b e giv en satis- fying the assumption. L et p O δ q δ P α 1 b e a co v er of X , and let p U ε q ε P α b e deﬁned by U ε  O δ , if f p δ q  ε , a nd arbitra rily , if ε is not in the image of f . The deﬁnition is w ell p osed, since f is injectiv e. Let α 2 b e the order t yp e of f p α 1 q p U ε q ε P f p α 1 q is still a co v er of X , hence, by r β , α 2 s -compactness (whic h follo ws fr o m r β , α s -compactness, b y Prop osition 2.3(1)), and b y Re- mark 2.2 , there is K  α of order t ype  β and suc h that p U ε q ε P K still co v ers X . If w e put H  f  1 p K q , then, by assu mption, H has order t yp e  β 1 ; moreo v er, p O δ q δ P H is a cov er of X , hence r β 1 , α 1 s - compactness is prov ed. In case τ is closed under unions, and f is not injectiv e, deﬁne U ε   f p δ q ε O δ , and the same arg umen t carries ov er.  In what fo llo ws , if not otherwise sp eciﬁed, the op eration  will de- note o r dinal s um . That is, α  β is the order type of the order obtained 10 P AOLO LI PP AR INI b y at t a c hing a copy of β “at the top” of α . Similarly ,  denotes or d inal pr o duct . The next corollary provid es a sample of results that can b e prov ed ab out t he relationship b et w een r β , α s -compactness, and r β 1 , α 1 s -com- pactness, for v arious ordinals. Most of them shall b e used in the rest of the pap er. Corollary 2.6. Sup ose that α , β and γ ar e nonzer o o r dinals, a nd λ , and ν ar e c ar dinals. (1) If β ¤ α , and α is i n ﬁnite, then r β , α s -c om p actness implies r β , α  1 s -c om p actness, he n c e also r β , α  n s -c om p actness, for e ach n  ω . (2) If either γ o r α is inﬁnite, then r γ  α, γ  α s -c om p actness implies r γ  α  α, γ  α  α s -c om p actness, henc e also r γ  α  n, γ  α  n s -c om p actness, for e ach n  ω . (3) If β ¤ α , α is inﬁnite, an d λ  cf α , then r β , α s -c om p actness implies r β , α  λ  ω q -c om p actness. (4) If β ¤ λ , then r β , λ s -c om p actness implies r β , λ  q -c om p actness. (5) If β ¤ α  λ , and either cf α ¡ λ , or α c an b e written as a limit of or dinals o f c oﬁnality ¡ λ , then r β , α  λ s -c om p actness implies r β , α  λ  q -c om p actness. Supp ose further that τ is close d under unions. T hen: (6) r α, α s -c om p actness implies r β  α, β  α s -c om p actness. (7) r α, α s -c om p actness implies r β  α, β  α s -c om p actness. (8) If cf α  ν is i n ﬁnite, then r ν, ν s -c om p actness implies r α, α s - c om p actness. Pr o of. (1) In view of Prop osition 2.3( 1), and since β ¤ α , r β , α s -com- pactness implies r α, α s -compactness. In view of Prop osition 2.3(3), it is then enough to show that r α, α s -compactness implies r α  1 , α  1 s - compactness. The latter is pro v ed b y applying Prop osition 2.5 to the f unction f : α  1 Ñ α deﬁned as follo ws. f p ε q  $ ' & ' % 0 if ε  α, ε  1 if ε  ω , ε if ω ¤ ε  α. (2) If α is ﬁnite, then γ is inﬁnite, and the result f ollo ws from (1). Otherwise, suppo se that α  α 1  n , with α 1 limit and n  ω . Th us γ  α  α  γ  α 1  α 1  n . Consider the follo wing function ORDINA L COMP ACT NESS 11 f : γ  α  α Ñ γ  α . f p ε q  $ ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' % ε if ε  γ , γ  2 m if ε  γ  m , with m P ω , γ  α 2  2 m if ε  γ  α 2  m , with α 2 limit  α 1 , m P ω , γ  2 m  1 if ε  γ  α 1  m , with m P ω , γ  α 2  2 m  1 if ε  γ  α 1  α 2  m , with α 2 limit  α 1 , m P ω , γ  α 1  m if ε  γ  α 1  α 1  m , with m  n. It is easy to see that f is injective . Supp ose that K  γ  α  γ  α 1  n , and K has o rder t yp e  γ  α . Then either (a) K X r γ  α 1 , γ  α 1  n q has order t yp e  n , or (b) K X r γ , γ  α 1 q has order t ype α   α 1 , or (c) K X γ has o rder t yp e γ   γ . If (a) holds, then f  1 pr γ  α 1 , γ  α 1  n qq has order t yp e  n , hence f  1 p K q has order ty p e  γ  α 1  α 1  n  γ  α  α . In case (b), f  1 pr γ , γ  α 1 qq has or der t yp e ¤ α   α  , hence f  1 p K q has order t yp e ¤ γ  α   α   n , whic h is strictly smaller than γ  α 1  α 1  n , since α   α 1 . Finally , w e can supp ose that w e are in case (c), and b oth (a) and (b) fail. Since K has order t yp e  γ  α  γ  α 1  n and K X γ has order ty p e γ   γ , then γ   α  γ  α . This easily implies that γ   α  α  γ  α  α (for example, b y expres sing γ  , γ and α in Can tor normal form). Since f is injectiv e a nd, restricted to γ , is the iden tit y , then f  1 p K q has order t yp e ¤ γ   α  α  γ  α  α . W e hav e pr ov ed that f  1 p K q has order type  γ  α  α in all cases, hence Prop osition 2.5 can b e applied. (3) If cf α  1, this follows from (1), hence let us supp o se that cf α ¥ ω . By Prop osition 2.3, it is enough to pr ov e that if δ  λ  ω , then r α, α s - compactness implies r α  δ , α  δ s -compactness. Reﬁning further, it is enough to prov e that (*) if δ ¤ λ , then r α, α s -compactness implies r α  δ , α  δ s -compactness, since then r α, α s -compactness implies r α  λ, α  λ s -compactness, and then we can pro ceed inductiv ely , by applying the result with α  λ in place of α , and then with α  λ  λ in place of α , and so on. Hence, supp ose that δ ¤ λ  cf α , and that r α, α s -compactness holds. If δ  α ¡ α , then necessarily δ  cf α and α  p cf α q  m , for some m  ω , and (*) follows from (2) with γ  0. Otherwise, δ  α  α , 12 P AOLO LI PP AR INI hence w e can deﬁne the follow ing injectiv e function f : α  δ Ñ α . f p ε q  # δ  ε if ε  α, η if ε  α  η , for η  δ . No w, if K  α has order type ζ  α , then f  1 p K q has order t yp e ¤ ζ  δ , whic h is necessarily  α  δ , since δ ¤ cf α . Hence Prop osition 2.5 can b e applied in order to get (*). (4) Again by Prop osition 2.3, it is enough to prov e that r λ, λ s -com- pactness implies r α, α s -compactness, for ev ery α with | α |  λ . This is accomplished b y Pro p osition 2.5, letting f b e an y injection from α to λ . (5) As ab o v e, it is suﬃcien t to pro v e that r α  λ, α  λ s -compactness implies r α  γ , α  γ s -compactness, for ev ery γ with | γ |  λ . Let g b e an y injection from γ to λ , and apply Prop osition 2.5 to the fo llo wing function f : α  γ Ñ α  λ . f p ε q  # ε if ε  α, α  g p η q if ε  α  η , with η  γ . If K  α  λ has order type  α  λ , then either K X α has o r der t yp e  α , or K X r α, α  λ q has order type  λ . In the la t ter case, and since λ is a cardinal, w e ha v e that f  1 p K q has order t ype ¤ α  γ 1 , for some γ 1 with | γ 1 |  λ , hence f  1 p K q has order type  α  γ , since | γ |  λ . On the o t her hand, if K X α has o r der t yp e  α , then f  1 p K q X α has order t yp e  α , since f is the iden tit y on α . The assumptions on α , and | γ |  λ then imply that f  1 p K q has order type  α  γ . (6) Apply the la st stateme n t in Prop osition 2.5 to the function f : β  α Ñ α deﬁned by f p ε q  # 0 if ε  β , η if ε  β  η , with η  α. (7) Apply the la st stateme n t in Prop osition 2.5 to the function f : β  α Ñ α deﬁned by f p ε q  ζ if ε  β  ζ  η , for some η  β . (8) Let p γ η q η P ν b e a sequence coﬁnal in α o f order t yp e ν . Deﬁne f : α Ñ ν b y f p ε q  inf t η P ν | ε  γ η u , and apply Prop osition 2.5.  Example 2.7 . As suggested by Corolla ry 2.6 (6)- (8), the relationships b et w een v ar io us ordinal compactness prop erties c hange according to whether τ is required or no t to b e closed under unio ns. F or example, if λ ¡ µ are inﬁnite cardinals, then ev ery r µ, µ s -compact top ological space is r λ  µ, λ  µ s -compact, b y Corollary 2.6(6). On the other hand, ORDINA L COMP ACT NESS 13 if X  p λ  µ, τ q , where τ  tr 0 , β q | β P λ u Y tr λ, λ  γ q | γ P µ u , then X is trivially r µ, µ s -compact (since it has no cov er of cardinality µ ), but it is not r λ  µ, λ  µ s -compact. This is a n example of a more general fact: see Corollary 5.6. See a lso Example 4.4. W e shall see in Sections 3 and 5 t ha t r β , α s -compactness is v ery far f rom b eing a trivial notion. Ho w ev er, Corollary 2.6(4) implies that r β , α s -compactness b ecomes partly trivial for in terv als containing a car- dinal. Corollary 2.8. If α is inﬁnite, a nd β ¤ | α | , then the fol lowi ng pr op- erties a r e e quivalent. (1) r β , | α |s -c om p actness. (2) r β , | α |  q -c om p actness. (3) r β , α s -c om p actness. In p articular, if µ ¤ λ ar e inﬁ n ite c ar d i n als, then r µ, λ s -c om p actness is e quivalen t to r µ, λ  q -c om p actness. Pr o of. (1) ñ (2) is from Corollary 2.6(4). (2) ñ (3) and (3) ñ (1) are immediate from Pro p osition 2.3(1).  In particular, “initial α - compactness ”, that is, r ω , α s -compactness, do es become tr ivial, in the sense that it actually reduces to cardinal compactness, in fact, to r ω , | α |s -compactness. The next Lemma gives a somewhat useful equiv alen t fo rm ulation of r β , α s -compactness. It states that it is enough to take into account only co v ers whic h are made o f “irredundan t” elemen ts. Lemma 2.9. L e t X b e a nonem pty set, τ b e a n o nempty family of subsets o f X , and β , α b e nonzer o or dinals. Then p X , τ q is r β , α s -c om p act i f and only if the fo l lowing c ondition holds. Whenever α  ¤ α , and p O δ q δ P α  is a se quenc e of memb ers of τ such that (1)  δ P α  O δ  X , and (2) for every δ  α  , O δ is not c o n taine d in  ε  δ O ε , then ther e is H  α  with or der typ e  β and such that  δ P H O δ  X . Pr o of. The “only if ” part follo ws trivially from Prop osition 2.3(1). Con v ers ely , suppose that p O δ q δ P α is a co v er of X . Let K  t δ P α | O δ is not con tained in  ε  δ O ε u . Clearly , p O δ q δ P K is still a cov er of X . Let α  b e the order t yp e of K , and let f : α  Ñ K b e the order preserv- ing bijection. Applying the assumption to the sequence p O f p γ q q γ P α  , w e get H  α  with order t yp e  β , suc h that  γ P H O f p γ q  X . This 14 P AOLO LI PP AR INI means that p O δ q δ P f p H q is a co v er of X indexed by a set of o r der t ype  β . In particular, it is a sub co v er of p O δ q δ P α th us r β , α s -compactness is pro v ed.  3. First e xamples In this section w e provide many examples sho wing tha t ordinal com- pactness is not a “trivial” notion. In particular, it cannot b e reduced to cardinal compactness. W e also sho w that man y o f the results prov ed in Corolla ry 2.6 are the b est p ossible ones, in the g eneral case. On the con trary , w e shall sho w in Section 6 that certain results can b e impro v ed if w e just assume tha t w e are dealing with a T 1 top ological space. In subsection 3.1 we endow cardinals with sev eral top olog ies, and c haracterize exactly the ordinal compactness prop erties they share. Then in Subsection 3.2 w e giv e detailed results ab out compactness prop erties of disjoin t unions, and show that taking disjoint unions is a v ery ﬂexible w a y to get more counterex amples. Examples of a diﬀeren t kind shall b e presen ted in Section 5. Finally , in Subsection 3.3 w e discuss the tec hnical notio n o f a shifted sum of t w o ordinals, introduced in connection with compactness prop- erties of disjoin t unions. 3.1. Basic examples. Deﬁnition 3.1. W e shall endow cardinals with sev eral top olo g ies. As usual, the discr ete top ology d (on an y set) is the trivial to p olo g y in whic h ev ery subset is op en. The initial interval top olo gy iit on some cardinal λ is the top olo g y whose op en sets are the interv als of the f o rm r 0 , β q , with β ¤ λ . The or der top olo gy ord on some cardinal λ is the more usual top ology; a base for this top ology is g iv en b y the inte rv als p α, β q ( α  β ¤ λ ), and r 0 , β q ( β ¤ λ ). Examples 3 .2 . Let λ b e an y cardinal, and κ b e an inﬁnite regular car- dinal. (1) p λ, d q is r λ  , 8q -compact, and no t r α, α s -compact, for ev ery nonzero α  λ  . (2) p κ, iit q is not r κ, κ s -compact, but it is r κ  1 , 8q -compact, and r 2 , κ q -compact. (3) If κ ¡ ω , then p κ, ord q is a normal top ological space whic h is r κ  ω , 8q -compact, r ω , κ q -compact, and not r κ  n, κ  n s - compact, for eac h n P ω . ORDINA L COMP ACT NESS 15 Pr o of. (1) is trivial. (2) The sequence r 0 , β q β  κ itself prov es r κ, κ s -incompactness, since κ is an inﬁnite regula r cardina l. On the ot her hand, let p O δ q δ P α b e a co v er of p κ, iit q . If O δ  κ , for some δ P α , then clearly t O δ u itself is a one-elemen t sub co v er. Supp ose otherwise. Since κ is regular, then necessarily α ¥ κ , and our aim is to extract a sub co v er of order ty p e ¤ κ . In fact, the sub co v er will turn out to b e o f order t yp e exactly κ . By Lemma 2.9, the result follows from t he particular case in whic h the co v er p O δ q δ P α has the additional prop ert y that, f o r ev ery δ  α , O δ is no t con tained in  ε  δ O ε . Supp ose that the ab o v e condition is satisﬁed. Since each O δ has the form r 0 , β δ q , for some β δ  κ , then, b y the ab o v e condition, β δ  β δ 1 , for all pairs δ  δ 1  α . Since p O δ q δ P α is a co v er of κ , then sup δ  α β δ  κ . Thus , the sequence p β δ q δ  α is strictly increasing, and coﬁnal in κ , hence has order ty p e κ , since κ is a r egula r car dina l. (3) Let p O δ q δ P α b e a co v er of p κ, ord q . First, consider the case when some O ¯ δ con tains an interv a l of the form p ε, κ q , for some ε  κ . Since r 0 , ε s is compact, it is co v ered b y a ﬁnite n um ber of the O δ ’s. If we add O ¯ δ to these, w e get a ﬁnite sub co v er of κ , since κ  r 0 , ε s Y p ε, κ q , hence the conclusion holds in this case. So w e can supp ose that no O δ con tains an interv a l of the form p ε, κ q , th us necessarily α ¥ κ , since κ is regular. Since p O δ q δ P α is a cov er, and each O δ is a union of interv a ls, w e ha v e that, for ev ery β P κ , with β  0, there is an in terv al I β  p ε β , φ β q , with ε β  φ β  κ , suc h that β P I β , and I β  O δ p β q , for some δ p β q P α . F or ev ery nonzero β P κ , c hoo se some I β and some δ p β q P α as ab ov e. The function f : κ zt 0 u Ñ κ deﬁned b y f p β q  ε β is regressiv e , hence constant on a set S stationa ry in κ , say , f p β q  ¯ ε , for β P S . Let D  t δ P α | δ  δ p β q , for some β P S u . F or δ P D , let η δ  sup t η  κ | O δ  p ¯ ε, η qu , and let J δ  p ¯ ε, η δ q . Th us, O δ  J δ , for δ P D . Moreo v er, I β  J δ p β q , for β P S , since, if β P S , then p ¯ ε, φ β q  I β  O δ p β q . W e no w sho w that p J δ q δ P D is a co ver of p ¯ ε, κ q . Indeed, sinc e S is stationary , in particular, coﬁnal, then, for eve ry β 1 with ¯ ε  β 1  κ , there is β ¡ β 1 , suc h that β P S , thu s β 1 P I β , since ¯ ε  β 1  β P I β  p ε β , φ β q , hence β 1 P J δ p β q  I β . Since p ¯ ε, κ q is order-isomorphic to κ , and, thro ug h this isomorphism, the J δ ’s cor r espond to op en sets in the iit top ology , w e can apply (2) in order to get a subset E  D  α suc h that E has order ty p e ¤ κ , and p J δ q δ P E co v ers p ¯ ε, κ q . Hence also p O δ q δ P E co v ers p ¯ ε, κ q . 16 P AOLO LI PP AR INI Since κ  r 0 , ¯ ε s Y p ¯ ε, κ q , and r 0 , ¯ ε s is compact, it is enough to add to E a ﬁnite n um b er of eleme n ts from the original seq uence p O δ q δ P α , in order to get a co v er of the whole κ . Since w e ha v e a dded a ﬁnite n um ber of elemen ts to a sequenc e o f order t ype ¤ κ , we get a cov er of κ whic h has o r der t yp e  κ  ω , and whic h is a subsequence of the original sequence. Thus , we ha v e pro v ed r κ  ω , α s -compactness. In order to ﬁnish the pro of, w e hav e to show that, for eac h n P ω , p κ, ord q is not r κ  n, κ  n s -compact. An easy counterex ample is giv en b y the sequence p O δ q δ P κ  n deﬁned b y O δ  # r n, n  δ q if δ  κ, t m u if δ  κ  m , with m  n  The situation app ears in a clearer ligh t if w e intro duce an ordinal v ariant of the L indel¨ of num b er of a space. Deﬁnition 3.3. The Lindel¨ of or d i n al of p X , τ q is the smallest ordinal α suc h that p X , τ q is r α, 8q -compact. Compare the ab ov e deﬁnition with the classical notion of the Linde l ¨ of numb er of a top olo g ical space X , whic h is the smallest c ar dinal µ suc h that X is r µ  , 8q -compact (the Lindel¨ of numb er is a distinct notion from the Lindel¨ of  c ar dinal deﬁned in the intro duction.) Th us, the Lindel¨ of n um b er µ of X is determined by its Lindel¨ of ordinal α . Indeed, µ is t he predecessor of α , if α is a successor cardinal, and µ  | α | otherwise. On the other hand, in general, the Lindel¨ of ordinal cannot b e determined b y the Lindel¨ of num b ers, as show n by Example 3.2. Indeed, taking λ  κ regular and uncoun table, all the spaces in Examples 3.2 hav e Lindel¨ of nu m b er equal to κ , ho w ev er, their Lindel¨ of ordinals are, r espectiv ely , κ  , κ  1, and κ  ω . Other p ossibilities for the Lindel¨ of ordinal are presen ted in Examples 3 .10, 3.11 and 3.12. On the other hand, restrictions on t he p ossible v alues Lindel¨ of or dina ls can assume ar e giv en in Corolla ry 4 .8 fo r spaces o f small cardinalit y , and in Corollary 6.11 for T 1 spaces. R emark 3.4 . Examples 3.2 also show that o rdinal compactness cannot b e determined exclusiv ely b y the cardina l compactness prop erties en- jo y ed by some space. F or example, X 1  p ω , iit q is r ω  1 , 8q -compact, hence, by Prop osition 2.3(1), it is r α, α s -compact, for ev ery or dina l α ¡ ω . On the other hand, X 2  p ω , d q is r ω 1 , 8q -compact, but not r α, α s -compact, for ev ery coun table ordinal α . Th us, X 1 and X 2 are r λ, µ s -compact exactly fo r the same pairs of inﬁnite cardinals λ and µ , but there are man y ordinals α for which X 1 is r α, α s -compact, but X 2 is not. ORDINA L COMP ACT NESS 17 Example 3.11 b elow furnishes tw o normal top ological spaces which are r λ, µ s -compact exactly f o r the same pairs of cardinals λ and µ , no matter whether ﬁnite or inﬁnite, but not r α, α s -compact for the same ordinals. 3.2. Disjoin t unions. In order to reﬁne Examples 3 .2, w e need some deﬁnitions. Deﬁnition 3.5. If X 1 and X 2 are sets, with τ 1 , τ 2 resp ectiv e fa milies of subsets, the disjoint union p X 1  Y X 2 , τ q of p X 1 , τ 1 q and p X 2 , τ 2 q is a set X 1  Y X 2 obtained by t a king the union of disjoin ted copies of X 1 and X 2 , with τ b eing the fa mily of a ll subsets of X 1  Y X 2 whic h either b elong to (the copy o f ) τ 1 , o r b elong to τ 2 , or are the union of a set in τ 1 and a set in τ 2 . Of course, in the case when X 1 and X 2 are top ological spaces, w e get back the usual notion of disjoint union in the t o po lo gical sense. Deﬁnition 3.6. If α and β are or dinals, w e say that some o r dinal γ is a shif te d sum of α and β if and only if γ  I Y J , fo r some (not necessarily disjoin t) subsets I , J  γ such that I has or der type α and I has order type β . T rivially , b oth α  β and β  α are shifted sums of α and β . The (Hessen b erg) na tural sum α ` β is the largest p ossible shifted sum of α and β . This is immediate from [Car, Theorem 1, I, I I], where the Hessen b erg natural sum is denoted b y σ p α, β q , and follo ws also from Prop osition 3.16 b elo w. Ho w ev er, there a re other p ossibilities for shifted sums. F o r example, ω 1  ω is a shifted sum of ω 1 and ω  ω . A quite inv olve d fo rm ula f or determining all the p ossible shifted sums of α and β shall b e obtained in Prop osition 3.16, by expressing ordinals in additiv e normal fo rm. The complication arises from t he fact that, say , though b oth ω 3  ω  1 and ω 3  ω 2  1 are shifted sums o f α  ω 3  ω and β  ω 2  1, on the con trary ω 3  1 is not a shifted sum of α and β . If α and β are ordinals, w e denote b y α   β the smallest ordinal δ larger than all the shifted sums of α 1 and β 1 , for α 1  α and β 1  β . Alternativ ely , α   β can b e deﬁned as sup α 1  α,β 1  β α 1 ` β 1  1. W e shall also need the f o llo wing lemma. Lemma 3.7. S upp ose that γ is a shifte d s um of α and β , that is, γ  I Y J , w ith I having or der typ e α and J having or der typ e β . Then the fol lowing additional pr op erty is satisﬁe d. Whe n ever I   I has stil l or der typ e α , and J   J has stil l or der typ e β , then I  Y J  has stil l or de r typ e γ . 18 P AOLO LI PP AR INI Pr o of. Express γ in a dditiv e normal form as γ  ω η h  ω η h  1      ω η 1  ω η 0 , for some in teger h ¥ 0, and ordinals η h ¥ η h  1 ¥    ¥ η 1 ¥ η 0 . Put γ h  1  0 and, for i  0 , . . . , h , put γ i  ω η h  ω η h  1      ω η i  1  ω η i . Consider the interv als K i  r γ i  1 , γ i q , for i  0 , . . . , h . Clearly , each K i has order type ω η i . Moreo v er, γ is t he disjoin t union of the K i ’s. Fix some ¯ ı . Since γ  I Y J , then K ¯ ı  p I X K ¯ ı q Y p J X K ¯ ı q . Since K ¯ ı has order t ype ω η ¯ ı , then, b y an easy prop ert y of suc h expo nen ts, either I X K ¯ ı or J X K ¯ ı has order t ype ω η ¯ ı (this is similar to, e. g., Hilfssatz 1 in [L¨ a]). Supp ose that, sa y , I ¯ ı  I X K ¯ ı has order type ω η ¯ ı . Let I   I X p K h Y    Y K ¯ ı  1 q , and I   I X p K ¯ ı  1 Y    Y K 0 q , and let α  , α  b e their resp ectiv e order t yp es. Since γ is the union of the K i ’s, then I  I  Y I ¯ ı Y I  . Because of the relative wa y the elemen ts of the K i ’s a re ordered in γ , w e ha v e tha t α  α   ω η ¯ ı  α  . Notice t ha t α   ω η ¯ ı  1 , since the o r der ty p e o f K i  1 Y    Y K 0 is ω η i  1      ω η 0  ω η ¯ ı  ω  ω η ¯ ı  1 . Since I   I , then the order types of, resp ectiv ely , I  X I  , I  X I ¯ ı , and I  X I  are ¤ than, resp ectiv ely , α  , ω η ¯ ı , and α  . How ever, since b oth I  and I ha v e order ty p e α , then necessarily I  X I ¯ ı  I  X K ¯ ı has order t yp e ω η ¯ ı , since otherwise the order t ype of I  w ould b e strictly smaller tha n α  α   ω η ¯ ı  α  , since, as w e men tioned, α   ω η ¯ ı  1 . In a similar w a y , if J X K ¯ ı has order ty p e ω η ¯ ı , then also J  X K ¯ ı has order type ω η ¯ ı . Since the ab ov e arg umen t works for each i , w e get that, for each i  0 , . . . , h , either I  X K i or J  X K i con tribute to K i with order ty p e ω η i , that is, p I  Y J  q X K i has order type ω η i . This, together with the deﬁnition of the K i ’s, implies that I  Y J  has order t yp e γ .  In the next lemma we characterize the compactness prop erties of disjoin t unions. T he lemma has not the most g eneral form p ossible, but it is quite go o d for o ur purp oses. Lemma 3.8. Assume the notation in Deﬁn i tion s 3.5 an d 3.6. (1) Supp ose that X 1 is not r α, α s -c om p act, and X 2 is not r β , β s - c om p act. If γ is a shifte d sum of α an d β , then X 1  Y X 2 is not r γ , γ s - c om p act. In p a rticular, X 1  Y X 2 is neither r α  β , α  β s -c om p act, nor r β  α, β  α s -c om p act, nor r α ` β , α ` β s -c om p act. ORDINA L COMP ACT NESS 19 (2) If X 1 is r β 1 , α s -c om p act, a nd X 2 is r β 2 , α s -c om p act, then X 1  Y X 2 is r β 1   β 2 , α s -c om p act. Pr o of. (1) Represen t γ as I Y J as in the deﬁnition of a shifted sum, with I of order t yp e α and J of order type β , and let f 1 : I Ñ α and f 2 : J Ñ β b e the order preserving bijections. Let p O δ q δ P α b e a co v er of X 1 witnessing r α, α s -incompactness, and let p P ε q ε P β b e a cov er of X 2 witnessing r β , β s -incompactness. F or φ P γ , let Q φ  X 1  Y X 2 b e deﬁned b y Q φ  $ ' & ' % O δ if φ P I z J a nd δ  f 1 p φ q , P ε if φ P J z I and ε  f 2 p φ q , O δ Y P ε if φ P I Y J , δ  f 1 p φ q , and ε  f 2 p φ q . By the deﬁnition of disjoint union, p Q φ q φ P γ is a co v er of X 1  Y X 2 with elemen ts in τ . Supp ose that H  γ , and that p Q φ q φ P H is still a cov er of X 1  Y X 2 . Then it is easy t o see t ha t p O δ q δ P f 1 p H X I q is a cov er of X 1 . Since p O δ q δ P α witnesses the r α, α s -incompactness of X 1 , then f 1 p H X I q has order t yp e α , hence also I   H X I ha s order t yp e α , since f 1 is an order preserving bijection. Similarly , J   H X J has o rder t yp e β . By Lemma 3.7, H  H X γ  H X p I Y J q  p H X I q Y p H X J q  I  Y J  has o rder type γ . Th us, p Q φ q φ P γ is a counterex ample to the r γ , γ s - compactness of X 1  Y X 2 . The last statemen t in (1) f o llo ws fro m the remarks in Deﬁnition 3.5. (2) Let p O δ q δ P α b e a co v er of X 1  Y X 2 . Let I b e the set o f all δ P α suc h that either O δ  P δ , for some P δ P τ 1 , or O δ  P δ Y Q δ , for some (unique pair) P δ P τ 1 and Q δ P τ 2 . Similarly , let J b e the set of all δ P α suc h tha t either O δ  Q δ , for some Q δ P τ 2 , or O δ  P δ Y Q δ , for some P δ P τ 1 and Q δ P τ 2 . Notice that I Y J  α , b ecause o f the deﬁnition of disjoin t union. Since p O δ q δ P α is a co v er of X 1  Y X 2 , then p P δ q δ P I is a cov er of X 1 , and, since I has or der t ype ¤ α , then, b y Remark 2.2, Prop osition 2.3(1), and the r β 1 , α s -compactness of X 1 , there is I   I suc h that I  has order ty p e β 1 1  β 1 , and p P δ q δ P I  is still a co v er of X 1 . Similarly , there is J   J suc h tha t J  has order ty p e β 1 2  β 2 , and p Q δ q δ P J  is a co v er of X 2 . Let γ b e the order t ype of I  Y J  . Then γ is a shifted sum of β 1 1 and β 1 2 , th us γ  β 1   β 2 . Since p O δ q δ P I  Y J  turns out to b e a co v er of X 1  Y X 2 , the conclusion follo ws.  Corollary 3.9. Supp ose that the Linde l ¨ of o r dinal of X 1 is β 1 , and that the Linde l ¨ of or dinal of X 2 is β 2 . Then the Lindel ¨ of or dinal of X 1  Y X 2 is β 1   β 2 . 20 P AOLO LI PP AR INI Pr o of. The Lindel¨ of ordinal of X 1  Y X 2 is ¤ β 1   β 2 , a s an immediate consequenc e of Lemma 3.8( 2 ). Hence, to pro v e equality , and in view of Prop osition 2.3(2), w e ha v e to sho w that, for ev ery γ  β 1   β 2 , there is γ 2 with γ ¤ γ 2  β 1   β 2 and suc h that X 1  Y X 2 is not r γ 2 , γ 2 s -compact. Let γ  β 1   β 2 . By the deﬁnition of β 1   β 2 , t here are β 1 1  β 1 , β 1 2  β 2 , and γ 1  β 1   β 2 suc h that γ ¤ γ 1 and γ 1 is a shifted sum of β 1 1 and β 1 2 . By assumption, X 1 is not r β 1 1 , 8q -compact, hence, b y Prop osition 2.3(2 ), there is β 2 1 ¥ β 1 1 suc h that X 1 is not r β 2 1 , β 2 1 s -compact, and necessarily β 2 1  β 1 . Similarly , there is β 2 2 suc h that X 2 is not r β 2 2 , β 2 2 s -compact, and β 1 2 ¤ β 2 2  β 2 . It follo ws trivially form the deﬁnition of a shifted sum, a nd from β 1 1 ¤ β 2 1 and β 1 2 ¤ β 2 2 , that there is some shifted sum γ 2 of β 2 1 and β 2 2 suc h that γ 1 ¤ γ 2 . By Lemma 3 .8 (1), X 1  Y X 2 is not r γ 2 , γ 2 s -compact. Since β 2 1  β 1 and β 2 2  β 2 , then γ 2  β 1   β 2 . Th us γ 2 is an ordinal as w an ted.  W e are now ready to presen t many improv ements of Examples 3.2. Examples 3.10 . Let κ b e a n inﬁnite regular cardinal, and n P ω , n ¥ 2. (1) If X is the disjoint union of tw o copies of κ with the initial in terv al top ology iit of D eﬁnition 3.1, then X is not r κ, κ s -com- pact, not r κ  1 , κ  1 s -compact, a nd not r κ  κ, κ  κ s -compact, but it is r κ  κ  1 , 8q -compact, r κ  2 , κ  κ q -compact, and r 3 , κ q -compact. Thus X has Lindel¨ of ordinal (Deﬁnition 3.3) κ  κ  1. (2) More generally , if X is the disjoin t union of n copies of κ with the initial in terv al top ology , t hen X is not r κ, κ s -compact, not r κ  κ, κ  κ s -compact, . . . , not r κ  n, κ  n s -compact, but it is r κ  n  1 , 8q -compact, r κ  n, κ  κ q -compact, r κ  κ  n  1 , κ  κ  κ q -compact, . . . , r κ  p n  1 q  2 , κ  n q -compact, and r n  1 , κ q -compact. Its Lindel¨ of ordinal is κ  n  1. Example 3.1 1 . Supp ose that κ is regular and ¡ ω , let X 1  p κ, ord q , and let X 2 b e the disjoin t union o f tw o copies of X 1 . Then b oth X 1 and X 2 are r µ, λ s -compact, for eve ry pair of inﬁnite cardinals µ and λ suc h that either κ  µ ¤ λ , or ω ¤ µ ¤ λ  κ ; furthermore, b oth X 1 and X 2 are not r κ, κ s -compact, and not r n, n s - compact, for ev ery p ositiv e integer n . Th us, X 1 and X 2 are r µ, λ s - compact exactly fo r the same pa ir s of car dina ls µ a nd λ , whether ﬁnite or not. Ho w ev er, X 1 is r κ  ω , 8q -compact, while X 2 is no t ev en r κ  κ, κ  κ s - compact. Actually , X 2 is not r κ  κ  n, κ  κ  n s -compact, fo r ev ery ORDINA L COMP ACT NESS 21 n  ω , but it is r κ  ω , κ  κ q -compact, and r κ  κ  ω , 8q -compact. Its Lindel¨ of ordinal is κ  κ  ω . Example 3.12 . Supp ose that X 1 is a nonempt y set, a nd τ is a nonempt y family o f subsets o f X 1 . Supp ose that X 2 is a discrete top ological space of cardinalit y µ , and tha t X is the disjoin t union o f X 1 and X 2 . Then the follow ing statemen ts hold. (1) If X 1 is not r α, α s -compact, | β | ¤ µ , and γ is a shifted sum of α and β , then X is not r γ , γ s -compact. (2) If X 1 is r β , α s -compact, then X is r β   µ  , α s -compact. In particular, b y adding a discrete ﬁnite set to Example 3 .2(2), w e can get a r κ  m  1 , 8q -compact space whic h is not r κ  m, κ  m s - compact. Th us we can hav e κ  m  1, a s a Lindel¨ of ordinal of some space. In a similar w a y , b y starting with Example 3.10, we can hav e κ  n  m  1 as a Lindel¨ of ordinal. Pr o ofs. Almost ev erything in Examples 3.10, 3.1 1 and 3.12 f ollo ws from Prop osition 2.3, Examples 3.2 a nd Lemma 3.8. An exception is r κ  2 , κ  κ q -compactness in Example 3.1 0(1), whic h is prov ed as follows. Let X b e the disjoint union of t w o copies of p κ, iit q , and consider an ordinal- indexed cov er C of X . By Example 3.2(2), there is a subsequence of C whic h is a cov er of the ﬁrst copy o f p κ, iit q and either has order t yp e κ , or consists of a single elemen t, that is, has order t yp e 1. Similarly , there is a subsequenc e of C which is a co v er of the second copy and has the same p ossible or der t yp es. By joining the ab o v e t w o partial sub cov ers, we get a cov er of the whole of X , whose order type is a shifted sum of β 1 and β 2 , where the p ossible v alues β 1 and β 2 are either κ or 1. An y suc h shifted sum, if  κ  κ , m ust necessarily b e ¤ κ  1, from whic h r κ  2 , α s -compactness follows , for ev ery α with κ  2 ¤ α  κ  κ . The pro ofs o f r κ  n, κ  κ q -compactness, r κ  κ  n  1 , κ  κ  κ q - compactness, . . . in 3.1 0(2), and of r κ  ω , κ  κ q -compactness in 3.11 are similar.  Man y other similar examples can b e obt a ined by combining in v ari- ous w a ys the examples in 3.2 with Lemma 3.8. F urther countere xam- ples can b e obtained b y a pplying disjoin t unions to the examples w e shall in tro duce in Deﬁnition 5.1. Example 3.13 . It is trivial to sho w that, f or µ ¤ λ inﬁnite car dina ls, the disjoin t union of tw o to po logical spaces is r µ, λ s -compact if and only if the t w o spaces are b oth r µ, λ s -compact (this also fo llo ws from Lemma 3.8). 22 P AOLO LI PP AR INI The space constructed in Example 3.11 sho ws that, for ordinals, the disjoin t union of tw o r β , α s -compact spaces is not necessarily r β , α s - compact. Just tak e α  β  κ  κ , for some regular κ ¡ ω , and consider the union of t w o disjoin t copies of p κ, ord q . One can also deal with the o b vious ly deﬁne d notio n of the disjoin t union of an inﬁnite family . It app ears to b e promising also the p os- sibilit y of considering a part ial compactiﬁcation of an inﬁnite disjoint union. This can b e a ccomplis hed as follo ws. Deﬁnition 3.14. Supp ose t ha t p X i q i P I is a family of nonempt y sets and, for eac h i P I , τ i is a nonempt y family of subsets of X i . Supp ose, for sak e of simplicit y , that eac h τ i con tains the empty set. The F r e chet d isjoint union p X , τ q of p X i , τ i q i P I is deﬁned as f o llo ws. Set theoretically , X  t x u  Y   i P I X i is the union of (disjoint copies) of the X i ’s, plus a new elemen t x whic h b elongs to no X i . The members of τ are those subsets O of X whic h ha v e one o f the follo wing t w o forms. O  ¤  i P I O i , where O i P τ i , for ev ery i P I , or O  t x u  Y ¤  i P I O i , where the O i ’s are suc h that, for some ﬁnite set F  I , it happ ens that O i P τ i , for i P F , and O i  X i , for i P I z F . The ab ov e deﬁnition app ears to be in teresting, in the presen t con- text, since, as in Example 3.13, r β , α s -compactness of a F rechet dis- join t union is not necessarily preserv ed. How ev er, (inﬁnite) cardinal compactness a nd man y other top olog ical prop erties are preserv ed, as asserted b y the next prop osition. Prop osition 3.15. If p X i q i P I is a family of top olo gic al sp ac es, then their F r e chet disjoint union X  t x u  Y   i P I X i is a top olo gic al sp ac e, and is T 0 , T 1 , Hausdorﬀ, r e gular, norma l , r λ, µ s -c om p act (for given inﬁnite c ar dinals λ and µ ) , has a b ase of c l o p en sets if a nd on ly if so is (has) e ac h X i . Pr o of. Straightforw a r d. W e shall commen t only on r egula r ity and nor - malit y . F or these, just o bserv e that if C is closed in X and C has nonempt y in tersection with inﬁnitely many X i ’s, then x P C .  Notice that the spaces in Examples 3.2(2) and 3.1 0 satisfy very few separation axioms. Indeed, just ass uming that X is a T 1 top ological ORDINA L COMP ACT NESS 23 space, it is imp ossible to construct similar coun terexamples. See Sec- tion 6. Curiously enough, Counterex ample 3.10 cannot b e generalized in a simple wa y in order to get a space X whic h is not r κ  κ, κ  κ s -compact, but whic h is, sa y , r κ  κ  κ, κ  κ  κ s -compact. Suc h a countere xample exists (Remark 5.5), but w e need a m uc h more inv olve d construction. Indeed, if X is suc h a coun terexample, then | X | ¡ κ , as w e shall sho w in the next section. 3.3. A note on shifted sums and mixed sums. W e now giv e the promised c haracterization of t hose ordina ls γ whic h can b e realized as a shifted sum of tw o ordinals α and β . Ev ery ordinal γ can b e expressed in a unique w a y in additive no rmal form as γ  ω η h  ω η h  1      ω η 1  ω η 0 , for some in teger h ¥ 0, and ordinals η h ¥ η h  1 ¥    ¥ η 1 ¥ η 0 . Hence to an y ordina l γ w e can uniquely asso ciate the ﬁnite string σ p γ q of ordinals in (not necessarily strictly) decreasing order η h η h  1 . . . η 1 η 0 . W e are allowing the empty string, which is asso ciated to the ordinal 0. T o ev ery string of ordinals σ  η h η h  1 . . . η 1 η 0 w e can asso ciate the ordinal δ p σ q  ω η h  ω η h  1      ω η 1  ω η 0 . W e ar e not necessarily assuming tha t the ordinals in σ are in decreasing order. Ho w ev er, an arbitrary string σ can b e r e duc e d to a string σ r whose elemen ts are in (no t necessarily strictly) decreasing o r der, by taking out from σ all those elemen ts whic h are follow ed fr om some strictly larger elemen t. Notice that, any w a y , δ p σ r q  δ p σ q , since, fo r example, ω ξ  ω ξ 1  ω ξ 1 , if ξ  ξ 1 . In particular, if γ  δ p σ q , then σ p γ q  σ r , since the corresp ondence b et w een ordinals and strings consisting of decreasing ordinals is bijectiv e. W e let  denote string juxtap osition. Prop osition 3.16. Supp ose that α , β , γ ar e or di n als, and σ p γ q  η h η h  1 . . . η 1 η 0 . Then the fol lowing c onditions ar e e quivalent. (1) γ is a shifte d sum of α and β . (2) Ther e ar e (p ossibly empty) string s σ h , . . . , σ 0 and σ 1 h , . . . , σ 1 0 such that (a) α  δ p σ h  σ h  1      σ 0 q , (b) β  δ p σ 1 h  σ 1 h  1      σ 1 0 q , (c) fo r e ach i  0 , . . . , h , either σ i  η i , or σ i is empty, or every element of σ i is  η i , (d) f o r e ac h i  0 , . . . , h , either σ 1 i  η i , or σ 1 i is empty, or every element of σ 1 i is  η i , 24 P AOLO LI PP AR INI (e) fo r e a ch i  0 , . . . , h , either σ i  η i , or σ 1 i  η i . (3) Sam e as (2) with c onditions (a ) an d (b) r e plac e d by (a 1 ) σ p α q  σ h  σ h  1      σ 0 , (b 1 ) σ p β q  σ 1 h  σ 1 h  1      σ 1 0 . Pr o of. F or i  0 , . . . , h , deﬁne the interv als K i as in the pro of of L emma 3.7. Recall tha t each K i has order ty p e ω η i , that γ is the disjoin t union of the K i ’s, and that, for ev ery i ¡ i 1 , eac h elemen t of K i precedes ev ery elemen t of K i 1 , in the ordering induced b y the ordering on γ . (1) ñ (2) By (1), γ  I Y J , for some I and J of order types, resp ectiv ely , α and β . F or i  0 , . . . , h , let α i b e the o rder type o f I X K i , th us α  α h      α 0 , by the ab ov e prop erties of the K i ’s. Put σ i  σ p α i q . Then (a) is satisﬁed, since δ p σ h      σ 0 q  δ p σ h q      δ p σ 0 q , and since δ p σ p ε qq  ε , for ev ery ordinal ε . Moreo v er, (c), to o, holds, since the order t yp e of α i is ¤ the order ty p e of K i , tha y is, ω η i . Similarly , letting β i b e the order t yp e of J X K i , and σ 1 i  σ p β i q , w e hav e that ( b) and (d) hold. F ina lly , as remark ed in the pro of of Lemma 3.7, since K i  p I X K i q Y p J X K i q has order t yp e ω η i , then either α i or β i has order t yp e ω η i , th us (e) holds. (2) ñ (3) Observ e t ha t p σ h      σ 0 q r  σ  h      σ  0 , for appropriate strings σ  i , suc h that eac h σ  i is a substring of σ i (ho w ev er, it is not necessarily the case that σ  i  σ s i ). Then, by the last remark b efore the statemen t of the prop osition, σ p α q  p σ h      σ 0 q r  σ  h      σ  0 . Moreo v er, if the σ i ’s satisfy (c), then also the σ  i ’s satisfy (c), since w e are just taking out elemen ts. F urthermore, if σ i  η i and (c) holds, then this o ccurrence of η i is not deleted in σ  i , since η i ¥ η i 1 , for i ¡ i 1 . By taking further strings σ 1 i suc h t ha t p σ 1 h      σ 1 0 q r  σ 1 h      σ 1 0 , and arguing as b efore, we get that the σ  i ’s and the σ 1 i ’s witness (3). (3) ñ (2) is tr ivial, since δ p σ p ε qq  ε , for eve ry o r dina l ε . (2) ñ (1) F or i  0 , . . . , h , put α i  δ p σ i q and β i  δ p σ 1 i q . By Clauses (c)-(d), α i and β i are b oth ¤ ω η i . Let I i b e the initial segmen t of K i of order t yp e α i , and J i b e the initial segmen t o f K i of order type β i . The deﬁnition is w ell p osed, since the order t yp e of K i is ω η i . By Clause (e), I i Y J i  K i . If w e put I  I 0 Y    Y I h and J  J 0 Y    Y J h , then I Y J  K 0 Y    Y K h  γ . Not ice that, by the pro p erties of the K i ’s, I has order type α h      α 0  δ p σ h q      δ p σ 0 q  δ p σ h      σ 0 q  α , b y Clause (a ). Similarly , by Clause (b), J has order type β , th us we are done.  Notice that, giv en α and β , there is o nly a ﬁnite n um b er of ordinals γ whic h are shifted sums of α and β . Indeed, by Prop osition 3.16, the elemen ts of σ p γ q are a (p ossibly prop er) subset of the union of the sets ORDINA L COMP ACT NESS 25 of the elemen ts of σ p α q and of σ p β q (coun ting multiplicitie s), and this can b e accomplished only in a ﬁnite n um b er of w a ys. On the ot her hand, given γ , it migh t b e the case that γ can b e realized in inﬁnitely man y w a ys as a shifted sum. F or example, for ev ery n  ω , ω ω  1 can b e realized as the shifted sum of ω ω and ω n  1. Notice that γ is the natural sum α ` β of α and β if and only if a represen t ation as in Prop osition 3.16 exists in suc h a w a y tha t , for eac h i  0 , . . . , h , either σ i  η i and σ 1 i is empt y , or σ 1 i  η i and σ i is empty . The notion of a shifted sum is related to a known similar notion, usually called mixed sum ( Mischsumme , [N, L ¨ a]). In our notation, γ is a mixe d sum of α and β if and only if and only γ can b e realized as a shifted sum of α and β as in Deﬁnition 3.6, with the additio nal assumption that I X J  H . Prop osition 3.17. Under the assumptions in Pr op osition 3.16, we have that γ is a mixe d sum of α and β if a n d only if Condition (2) (e quiva l e n tly, Condition (3)) in 3.16 holds with the fol lowing a dditional clause (f ) F or e ach i  0 , . . . , h , if η i  0 , then either σ i or σ 1 i is the empty string. Pr o of. If γ is a mixed sum of α and β , t hen, in particular, it is a shifted sum, hence the conditions in Prop osition 3.16(2)(3) hold. In order to pro v e (f ) , notice that, if γ is a mixed sum of α and β , and η i  0, then | K i |  1, hence either I i or J i is empt y , since, in the presen t situation, they are disjoint and con tained in K i , th us (f ) follows. It remains to sho w how to get disjoin t I i and J i , for eac h i , in the pro of o f (2 ) ñ (1) (hence w e get disjoin t I and J , since the K i ’s are pairwise disjoin t). If η i  0, this follo ws from Clause (f ). Otherwise , observ e that any set of order t yp e ω η i can alw a ys b e expresse d as the union o f tw o disjoin t subsets ha ving prescrib ed order ty p es α i and β i , pro vided that α i and β i are b oth ¤ ω η i , and their maxim um is ω η i .  A somewhat similar c haracterization of those ordinals γ whic h can b e expresse d as a mixed sum of α and β ha s b een given in [L¨ a]. Actually , [L¨ a] deals with mixed sums with p ossibly more than t w o summands. Also the results presen ted here can b e easily generalized to the case of more than tw o summands. W e lea v e this to the reader. W e no w discuss in more details the r elatio nship b et w een the notions of a shifted sum and of a mixed sum. It turns out that the only diﬀerence is made b y the “ ﬁnite tail” of γ , that is, if γ  γ   m , with 26 P AOLO LI PP AR INI γ  limit, then the wa y s γ  can b e realized as a shifted sum determine the w a ys γ can b e realized as a mixed sum. Corollary 3.18. L e t α , β , and γ b e or dinals. (1) Supp ose that γ is a limit or dinal. T hen γ is a mixe d sum o f α and β if and only if γ is a shi f te d sum of α and β (and, if this is the c ase, then either α or β is li m it, but no t ne c essarily b oth). (2) Mor e g e n er al ly, supp ose that γ  γ   m , with γ  limit, and ω ¡ m ¥ 0 . Then γ is a m ixe d sum o f α and β if and only if ther e ar e inte g e rs n, p ¥ 0 such that n  p  m , α has the form α   n , β has the form β   p , a nd γ  is a shifte d s um of α  and β  (one of α  and β  must thus b e limit, but not n e c essari l y b oth). Pr o of. (1) If γ is limit, and σ p γ q  η h η h  1 . . . η 1 η 0 , then a ll the η i ’s a re ¡ 0, thus Clause (f ) in Prop osition 3.17 is auto matically satisﬁed. (2) If γ  γ   m , then η i  0 exactly for i  0 , . . . , m  1, th us σ p γ  q  η h η h  1 . . . η m . The conclusion no w follows easily from (1) and Prop ositions 3.16 and 3 .1 7.  Notice that t he notions of a shifted sum and o f a mixed sum are dis- tinct. Indeed, it follows easily from Prop osition 3.16 that the smallest shifted sum of α and β is sup t α, β u . How ever, the smallest mixed sum of, sa y , ω  1 and ω  2 is ω  3 ¡ sup t ω  1 , ω  2 u . In general, as a corollary o f Prop osition 3.1 7, w e obtain a result b y Neumer [N]: for α  α   n and β  β   p , where α  and β  are limit ordinals, the smallest mixed sum of α and β is α   n  p , if α   β  , and sup t α, β u , if α   β  . 4. Some indispensability arguments and sp a ces of s mall cardinality As w e men tioned, a discrete space of cardina lity λ is not r α, α s - compact, for ev ery or dina l α of cardinality ¤ λ . In a more general w a y , w e can exhibit plent y of spaces whic h b eha v e as discrete spaces, that is, f o r whic h o r dina l (in- )compactnes s reduces to cardinal (in- )compactness. This is the theme of the ﬁrst prop ositions in the presen t section. Then we pro ceed to prov e a more sophisticated r esult, Theo- rem 4.5, which implies that, if we restrict ourselv es to spaces of cardi- nalit y κ , then r α, α s -compactness is equiv alent to r β , β s -compactness, for a large set of limit ordinals α and β of cardinalit y κ . In particu- lar, for coun table spaces, Corollary 4.7 shows that r α, α s -compactness b ecomes trivial abov e ω  ω . The ab ov e men tioned results imply that the relativ ely simple examples in tro duced in the previous section are ORDINA L COMP ACT NESS 27 really far from exhausting all p ossible kinds of coun terexamples. In- deed, further and more inv olved coun terexamples shall b e constructed in the next se ction. In fact, in the next section w e shall prov e some equiv alences whic h sho w that Prop osition 2.5 cannot b e improv ed. In order t o carry on the pro of of the next prop osition, w e need a deﬁnition. Deﬁnition 4.1. If p O δ q δ P α is a cov er of X , let us sa y that some O ¯ δ is i ndisp ensabl e if and only if ev ery sub co v er of p O δ q δ P α m ust contain O ¯ δ . Equiv a len t ly , O ¯ δ is indisp ensable if and o nly if there is x P O ¯ δ suc h that x R  δ P α,δ  ¯ δ O δ . F or example, if X is a top ological space with the discrete t opo logy , and p O δ q δ P α is a co v er o f X consisting of (all) singletons, then eac h elemen t of this cov er is indisp ensable. Prop osition 4.2. Supp ose that α is a non zer o or dinal, λ is an in- ﬁnite c ar dinal, and p X , τ q has some c over p O δ q δ P α having at le ast λ indisp ensable e lements. (1) If | α |  λ , then X is not r β , β s -c om p act, for every or dinal β with | β |  λ . (2) If τ is close d under unions, then X is not r β , β s -c om p act, for every nonzer o or dinal β with | β | ¤ λ . Pr o of. (1) Let | β |  λ . Rearrange the seque nce p O δ q δ P α as p O 1 ε q ε P β in suc h a w a y that, in this latter sequence, the subsequence of the indisp ensable elemen ts has or der type β . This is alwa ys p ossible, since λ is an inﬁnite cardinal, | β |  λ , and there are λ - man y indisp ensable elemen ts in the original sequence. F or example, if µ is the cardinality of the set of no n indispensable elemen ts (it may happ en that µ  0), c ho ose a subset Z  λ with | Z |  µ a nd suc h that | λ z Z |  λ , assign to non indispensable elemen ts only p ositions in Z , and assign all the other p ositions in β z Z to all indisp ensable elemen ts. Ev ery sub co v er of p O 1 ε q ε P β m ust contain all o f its indisp ensable el- emen ts, th us has order t ype β . This implies that X is not r β , β s - compact. (2) Let | β | ¤ λ , say | β |  ν . Consider a new cov er of X obtained b y c ho osing ν -many indisp ensable O δ ’s and joining all the remaining O δ ’s in to one of them (it is still in τ , since τ is closed under unions). If ν is ﬁnite, then the result is trivial. Otherwise, it is obta ined b y applying (1), with ν in place of λ , to this new cov er.  In Section 6 w e shall use argumen ts similar to those used in the pro of o f Prop osition 4.2 in order to pro v e results ab out compactness prop erties of T 1 spaces. 28 P AOLO LI PP AR INI Theorem 4.5 b elo w is a far more sophisticated result than Prop osi- tions 4.2. Recall tha t  and  denote, resp ective ly , ordinal sum and pro duct. Moreov er, also exp onen tiation, if not otherwise sp eciﬁed, will denote or d inal exp onentiation . Lemma 4.3. Supp ose that κ is a n inﬁnite r e gular c ar dinal and α is an or din a l of the form α 1  κ ε , for some or dinals α 1 ¥ 0 an d ε ¡ 1 such that ε is either a suc c essor or dinal, or cf ε  κ . Supp ose further that | X |  κ , and that p X , τ q is not r α, α s -c om p act. Then p X , τ q is not r α 1 , α 1 s -c om p act, for every lim i t or din al α 1 of the form α 1  κ  α 1 1 , for some α 1 1 ¡ 0 with | α 1 1 | ¤ κ . If, in addition, τ is close d under unions (in p articular, if τ is a top olo gy on X ), then p X , τ q is not r α 1 , α 1 s -c om p act, for every or dinal α 1 with | α 1 | ¤ κ . Pr o of. Supp ose tha t p O δ q δ P α is a counterex ample to r α, α s -compactness. In particular, for ev ery β  α , w e hav e  δ  β O δ  X prop erly . W e shall show a little more. Claim. F or every β  α , ther e a r e x P X z  δ  β O δ and γ x  α such that x R  δ ¥ γ x O δ (henc e, x P  β ¤ δ  γ x O δ , sinc e p O δ q δ P α is a c over of X ). Pr o of of the Claim. Supp ose by contradiction that the statemen t in t he claim fails. Then, fo r some giv e n β  α , w e ha v e that, for every x P X z  δ  β O δ , there ar e arbitrarily large indexes δ  α suc h tha t x P O δ . Fix some β as ab ov e, and en umerate the elemen t s in X z  δ  β O δ as p x γ q γ P κ 1 , with κ 1 ¤ κ (here w e are using the assumption that | X | ¤ κ ). W e shall deﬁne by tr a nsﬁnite induction a strictly increasing sequence p δ γ q γ P κ 1 suc h that x γ P O δ γ , for ev ery γ P κ 1 . First, c ho ose some δ 0  α suc h that x 0 P O δ 0 . Supp ose that γ  κ 1 , and that p δ γ 1 q γ 1  γ ha v e already been deﬁne d. Notice that, by the a ssu mption o n ε , the coﬁnality o f α  α 1  κ ε is κ . Since γ  κ 1 ¤ κ , and κ is regular, then sup γ 1  γ δ γ 1  α . Hence, b y the ﬁrst paragraph in the pr o o f, there is some δ γ ¡ sup γ 1  γ δ γ 1 suc h that x γ P O δ γ . Notice that t δ γ | γ P κ 1 u has order t ype κ 1 ¤ κ . Hence, if w e put D  r 0 , β q Y t δ γ | γ P κ 1 u , then D has order t ype ¤ β  κ 1 . Notice that β  κ 1  α , since α is of the form α 1  κ ε with ε ¡ 1, hence eac h ﬁnal subset of α has order ty p e κ ε ¡ κ . Ho w ev er, by construction,  δ P D O δ  X , hence w e hav e found a sub- co v er of p O δ q δ P α of order t ype  α , and this con tradicts the assumption that p O δ q δ P α witnesses the f ailure of r α, α s -compactness of X . W e hav e reac hed a contradiction, th us the claim is prov ed.  C l aim ORDINA L COMP ACT NESS 29 Pr o of of L emm a 4.3 ( c ontinue d) Now w e are g oing to construct by transﬁnite induction t w o sequenc es p x ξ q ξ P α 2 and p γ ξ q ξ P α 2 , for some or- dinal α 2 ¤ α , suc h that (1) x ξ b elongs to X , for every ξ  α 2 , (2) γ ξ 1  γ ξ  α , f o r ev ery ξ 1  ξ  α 2 , (3) γ 0  0, p γ ξ q ξ P α 2 is contin uous, and sup ξ P α 2 γ ξ  α , (4) x ξ P  t O δ | γ ξ ¤ δ  γ ξ  1 u , for ev ery ξ  α 2 , (5) x ξ R  t O δ | δ P r 0 , γ ξ q Y r γ ξ  1 α qu , for ev ery ξ  α 2 . Put γ 0  0. By applying the claim to β  γ 0  0, w e get x 0 P X and γ 1  α suc h that x 0 P  δ  γ 1 O δ and x 0 R  δ ¥ γ 1 O δ . Supp ose that x ξ and γ ξ  1 ha v e b een already deﬁned, for some ξ . Apply the claim to β  γ ξ  1 , in order to o bt a in x ξ  1 and γ ξ  2  α . No w suppo se that ξ is a limit ordinal, and that x ξ 1 and γ ξ 1 ha v e already b een deﬁned, for a ll ξ 1  ξ . If sup ξ 1  ξ γ ξ 1  α , tak e α 2  ξ , and terminate the induction. Otherwise, let γ ξ  sup ξ 1  ξ γ ξ 1 . Then apply the claim with β  γ ξ , in order to o bt a in x ξ and γ ξ  1 . It is immediate to show that the sequenc es constructed in suc h a w a y satisfy (1)-(5) ab ov e. Notice that, since | X |  κ , and X is not r α, α s -compact, then neces- sarily | α | ¤ κ . On the other hand, α ¥ κ , since α  α 1  κ ε , for ε ¡ 1. Hence | α |  κ . Moreov er, b y (2 ) a nd (3), and since cf α  κ , w e also get cf α 2  κ , th us | α 2 |  κ , since α 2 ¤ α . If w e assu me that τ is closed under unions, then the pro of can b e concluded in a rather simple w a y . Indeed, by letting U ξ   t O δ | γ ξ ¤ δ  γ ξ  1 u , fo r ξ  α 2 , we ha v e that x ξ P U η if and o nly if ξ  η . Th us p U ξ q ξ  α 2 is a co v er, b y (3), and since p O δ q δ P α is a co v er. Moreo v er, p U ξ q ξ  α 2 consists of | α 2 |  κ indisp ensable elemen ts, hence w e are done b y Prop osition 4.2(2). It remains to pro v e the theorem without the assumption that τ is closed under unions, and this in v olv es some t ec hnical computations. Hence, supp ose that α 1  κ  α 1 1 , for some α 1 1 ¡ 0 with | α 1 1 | ¤ κ . P artition α 2 in to α 1 1 -man y classes p Z η q η  α 1 1 , in such a wa y that | Z η |  κ , for ev ery η  α 1 1 . This is p ossible, since | α 2 |  κ , and | α 1 1 | ¤ κ . F or η  α 1 1 , put I η  r κ  η , κ  p η  1 qq , and W η   ξ P Z η r γ ξ , γ ξ  1 q . Notice that | W η |  κ , for ev ery η  α 1 1 . F or each η , let f η b e a bijection fr om I η on to W η . Notice tha t α 1   η  α 1 1 I η , and that eac h I η has order t yp e κ . Rearra nge the orig ina l cov er p O δ q δ P α as p O 1 ζ q ζ P α 1 according to the follow ing rule. If ζ P α 1 , then ζ P I η , for some unique η  α 1 1 ; then put O 1 ζ  O f η p ζ q . W e shall show that p O 1 ζ q ζ P α 1 witnesses r α 1 , α 1 s -incompactness of X . Indeed, since p Z η q η  α 1 1 is a partition of α 2 , then, by Condition (3) 30 P AOLO LI PP AR INI ab o v e, and by the deﬁnition of the W η ’s, we get that   η  α 1 1 W η  α . Since eac h f η is a bijection, and α 1    η  α 1 1 I η , w e get that p O 1 ζ q ζ P α 1 is actually a rearrangemen t of p O δ q δ P α , th us it is still a co v er of X . Let Y  α 1 , and supp ose t ha t p O 1 ζ q ζ P Y is a cov er of X . W e ha v e to sho w that Y has order t ype α 1 . It is enough to sho w that, for ev ery η  α 1 1 , | Y X I η |  κ , thus Y X I η and I η ha v e the same order type (  κ ). Hence Y and α 1 ha v e t he same order ty p e, since α 1   η  α 1 1 I η . So, ﬁx η  α 1 1 . F or ev ery ξ P Z η , by Condition (5 ) ab ov e, w e ha v e that x ξ R O δ , fo r ev ery δ P r 0 , γ ξ q Y r γ ξ  1 , α q . Since p O 1 ζ q ζ P Y is a cov er, there is ζ P Y suc h that x ξ P O 1 ζ . Ne cessarily , O 1 ζ  O δ , for some δ P r γ ξ , γ ξ  1 q , thus δ P W η , hence δ  f η p ζ q , and ζ P Y X I η . By construction, | Z η |  κ . Since, for ξ  ξ 1 , the in terv als r γ ξ , γ ξ  1 q and r γ ξ 1 , γ ξ 1  1 q are disjoin t, then, for eac h ξ P Z η , w e get a distinct δ P α , hence a distinct ζ P Y X I η , th us | Y X I η |  κ . Since the ab o v e argumen t w orks for eac h η  α 1 1 , w e get that p O 1 ζ q ζ P α 1 is indeed a countere xample to r α 1 , α 1 s -compactness.  Example 4.4 . If τ is not supp osed to b e closed under unions, the con- clusion in the second statemen t in Lemma 4 .3 might fail. Indeed, let κ b e an inﬁnite regular cardinals, let X  κ  κ , and let τ consist of the sets o f the form r κ  γ , κ  γ  δ s , for γ , δ  λ . Then p X , τ q is trivially not r κ  κ, κ  κ s -compact, but it is r κ  1 , κ  1 s -compact, since an y cov er of X alw a ys remains a co v er if w e take oﬀ an y single mem ber of the cov er. Actually , if | α |  κ , then X is r α, α s -compact if and only if α has not the form κ  α 1 , for some ordinal α 1 . The example also sho ws that the a ssu mption that τ is closed under unions is necessary in Condition (5) in Theorem 4.5 b elo w, as well a s in Condition (4) in Corollary 4.7. As a consequence of Lemma 4 .3, f o r spaces of cardinality κ , the theory of r α, α s -compactness b ecomes trivial on a large class o f limit ordinals, as explicitly stat ed in the next Theorem. More strikingly , for coun table spaces, the theory of r α, α s -compactness is non trivial only for ordinals ¤ ω  ω (Coro lla ry 4.7 b elo w). Theorem 4.5. If κ is a n inﬁn ite r e gular c ar dinal and | X |  κ , then the fo l low i n g c onditions ar e e quivalent. (1) X is r κ  κ, κ  κ s -c om p act. (2) X is r α, α s -c om p act, for some limit or dinal α of the fo rm α  κ  α 1 , for some α 1 ¡ 0 with | α 1 | ¤ κ . ORDINA L COMP ACT NESS 31 (3) X i s r α, α s -c om p act, for every or d i n al α of the form α 1  κ ε , for some or dinals α 1 ¥ 0 and ε ¡ 1 such that ε is either a suc c essor or din a l, or cf ε  κ . (4) X is r α, α  κ  ω q -c om p act, for every or dinal α o f the form α 1  κ ε , for some or dinals α 1 ¥ 0 and ε ¡ 1 such that ε is either a suc c essor or dinal, or cf ε  κ . If τ is close d unde r unions, then the pr e c e ding c onditions ar e also e quiv- alent to: (5) X is r α, α s -c om p act, for some nonzer o or dinal α such that | α | ¤ κ . Pr o of. (1) ñ (2) and (1) ñ (5) are trivial. (2) ñ (3) and, for τ closed under unions, (5) ñ (3) f o llo w fro m Lemma 4.3. (3) ñ (4) is fr om Corollary 2.6(3). (4) ñ (1) is immediate from Prop osition 2.3( 1 ), with α 1  0 and ε  2.  Corollary 4.6. Supp ose that κ is an inﬁnite r e gular c a r dinal, | X |  κ , and let A b e the set of al l or dinals α  κ  of the form κ  p α   n q , with cf α   κ and 0 ¤ n  ω . Then X is r α, α s -c om p act, for some α P A , if and only if X is r α, α s - c om p act, for al l α P A . Pr o of. Supp ose that X is r α 1 , α 1 s -compact, fo r some α 1 P A . Since α 1 is of t he f o rm given in Clause 4.5(2 ) , then all t he equiv alent conditions in Theorem 4.5 hold. No w let α  κ  p α   n q P A b e arbitrary . Since cf α   κ , then α   κ  α  , where α  is either successor or has coﬁnality κ itself. In b oth cases, α  κ  p α   n q is of the fo r m α 1  κ ε  κ  n , with ε ¡ 1 either successor, or of coﬁnalit y κ . Th us, X is r α, α s -compact, in force of Clause 4.5(4) and of Prop osition 2.3(1).  Corollary 4.7. If | X |  ω , then the fol lowing c onditions ar e e quiva- lent. (1) X is r ω  ω , ω  ω s -c om p act. (2) X is r α, α s -c om p act, for some c ountable limit or din a l α . (3) X is r ω  ω , 8q -c om p act. If τ is close d unde r unions, then the pr e c e ding c onditions ar e also e quiv- alent to: (4) X is r α, α s -c om p act, for some non zer o or d i n al α  ω 1 . Pr o of. The equiv alence of (1 ) , (2 ), and (4) is a particular case of The- orem 4.5 (Conditions (1 ) , (2) and (5) there). 32 P AOLO LI PP AR INI (3) ñ (1) is immediate from Prop osition 2.3. In order to ﬁnish the pro of, supp ose that (2) holds. Then, b y Theo- rem 4.5 (2) ñ (4), X is r δ , δ s -compact, for ev ery ordinal δ of the form α 1  ω ε  m , for ε ¡ 1, that is, for ev ery countable ordinal δ ¥ ω  ω . Since X , b eing coun table, is trivially r δ , δ s -compact for ev ery uncoun t- able δ , w e g et r ω  ω , 8q -compactness from Prop osition 2.3 (2). Hence (3) holds.  A result similar to Coro llary 4.7 holds for T 1 spaces (of a r bitrary cardinalit y): see Corolla ry 6 .8. Corollary 4.8. If | X |  ω , then the Lindel¨ of or dinal of X is either ω 1 , or is ¤ ω  ω . Mor e gener al ly, if κ is r e gular, and | X |  κ , then the Lindel¨ of or d i n al of X c an not ha ve the form α 1  κ ε  γ , with 0  γ  κ  ω , and ε ¡ 1 such that ε is either a suc c essor or dinal, or cf ε  κ . Pr o of. The ﬁrst statemen t is immediate from Corollary 4.7 (2) ñ (3). As f or the second statemen t, if the Lindel¨ of ordinal of X is  κ  , then X is r α, α s -compact, fo r some α as in Item (2) in Theorem 4.5. The conclusion now follows fr om Prop osition 2.3 and Item (4) in Theorem 4.5.  5. An exa ct characteriza tion of transfer proper ties In this section we introduce some further examples, more in v olv e d than those presen ted in Examples 3 .2. This is necessary in order to a v oid the limitat io ns giv en b y Theorem 4.5 and Coro lla ries 4.6 and 4.7. The examples in tro duced in this section are optimal, in the sense that they prov ide an exact c haracterization of those ordinals α and β suc h that r α, α s -compactness implies r β , β s -compactness. Deﬁnitions 5.1. As usual, w e denote b y α 2 the set of all the functions from α to 2  t 0 , 1 u . If f P α 2, the supp ort of f is t δ P α | f p δ q  1 u . F or nonzero ordinals β ¤ α , w e no w deﬁne S β p α q  t f P α 2 | the supp ort of f has order ty p e  β u . S β p α q is in a one-to o ne corresp ondence, via c haracteristic functions, with the set of a ll subsets of α which ha v e order type  β . The S in our notation is a reminder for Subset . How eve r, in the presen t note, w e shall mainly deal with elemen ts of α 2, rather than with subsets of α , since it will b e more con v enien t for our purp oses. W e shall mainly deal with the case β  α , and w e shall consider v arious families of subsets of S β p α q . ORDINA L COMP ACT NESS 33 W e put X p β , α q  p S β p α q , τ 0 q , where the elemen ts of τ 0 are all the subsets of S β p α q ha ving the f orm Z p ε q  t f P S β p α q | f p ε q  0 u , ε v arying in α . W e also let X U p β , α q  p S β p α q , τ U q , where τ U is the smallest fa mily of subsets of S β p α q whic h con tains τ 0 ab o v e, and is closed under unions. In other w ords, a generic elemen t o f τ U has the form  ε P H Z p ε q  t f P S β p α q | f p ε q  0, for some ε P H u , for some H  α . F or α ¤ 2 a nd β ¡ 1, neither τ 0 nor τ U are top ologies, since they are not closed under ﬁnite inte rsections. How eve r, if w e take the closure of τ U under ﬁnite intersec tions, w e do g et a top ology τ on S β p α q . F or ε ε ε  t ε 0 , ε 1 , . . . , ε n  1 u P S n  1 p α q , let Z p ε ε ε q  Z p ε 0 , ε 1 , . . . , ε n  1 q  Z p ε 0 q X Z p ε 1 q X    X Z p ε n  1 q  t f P S β p α q | f p ε 0 q  f p ε 1 q      f p ε n  1 q  0 u . Members o f τ ha v e t hen t he form  ε ε ε P H Z p ε ε ε q , H v arying a mong the subsets of S ω p α q . W e let X τ p β , α q  p S β p α q , τ q . The a bov e top ology τ is T 0 , but not ev en T 1 . A top olog y satisfying stronger separation a xioms can b e in tro duced as follow s. X T p β , α q  p S β p α q , τ T q , where τ T is the ( T ychonoﬀ ) top ology in- herited b y the pro duct top ology on α 2, where 2 is giv en the discrete top ology . Notice that X T p β , α q inherits from α 2 also the structure of a top ological group, W e shall write X p β q in place of X p β , β q , and similarly for X U p β q , X τ p β q , and X T p β q . The subscript τ is a reminder for top olo gy , the subscript U is a reminder for (closed under) Unions , and the subscript T is a reminder f or T ychonoﬀ . R emark 5.2 . Similar constructions, when restricted to cardinal num- b ers, ha v e sometimes b een considered in the literature. See, e. g., [AB, Example 4.1], [Li1] and [Ste, Example 4.2]. Lemma 5.3. Supp ose 0  β ¤ α , and assume the notations in Deﬁ- nition 5.1. If H  α , then the s e quenc e p Z p ε qq ε P H is a c over of X p β , α q if and only if H has or der typ e ¥ β . In p articular, X p β , α q is not r β , β s -c om - p act, henc e neither X U p β , α q , n or X τ p β , α q , n or X T p β , α q ar e r β , β s - c om p act. Pr o of. If H has order type  β , deﬁne f : α Ñ 2 by f p δ q  1 if and only if δ P H . Then f P X p β , α q , but f b elongs to no Z p ε q ( ε P H ). On the con trary , supp ose b y contradiction that H has order ty p e ¥ β , but t here is f P X p β , α q suc h that f b elongs to no Z p ε q ( ε P H ). If f R Z p ε q , then f p ε q  1, th us the supp ort of f con tains H , whic h has order t yp e ¥ β , and this contradicts f P X p β , α q . In order to sho w that X p β , α q is not r β , β s -compact, it is enough to c ho ose some H  α of order type β . Then, b y ab o v e, p Z p ε qq ε P H is a 34 P AOLO LI PP AR INI co v er of X p β , α q , but if K  H has order t ype  β , then p Z p ε qq ε P K is not a co v er of X p β , α q . The same argument w orks for X U p β , α q , X τ p β , α q , and X T p β , α q .  Theorem 5.4. L e t α and β b e non zer o or dinals, and assume the no- tations in Deﬁnition 5.1. Th e n the fo l low i n g c onditions ar e e quivalent. (a) X p β q is not r α, α s -c om p act. (b) Ther e exists an inje ctive function f : β Ñ α such that, for every K  α with o r der typ e  α , it happ ens that f  1 p K q has or der typ e  β . (c) F or arbitr ary p X , τ q , r α, α s -c om p actness implies r β , β s -c om p act- ness. Pr o of. (a) ñ (b). Supp ose that (a) holds. Then X p β q has a co v er p O δ q δ P α suc h that, whenev er H  α has order type  α , then p O δ q δ P H is no t a co v er o f X p β q . By Lemma 2.9, w e can suppo se that O δ  O δ 1 , for δ  δ 1 P α . Because of the deﬁnition o f τ 0 , for each δ P α , there is ε P β suc h tha t O δ  Z p ε q . Let W  t ε P β | Z p ε q  O δ , for some δ P α u  β . Since p O δ q δ P α is a co v er o f X p β q , t hen also p Z p ε qq ε P W is a co v er of X p β q . By Lemma 5.3, W has order ty p e β . Let g : W Ñ α b e deﬁned b y g p ε q  δ if and only if Z p ε q  O δ . Such a δ exists b ecause o f the deﬁnition of W , a nd is unique b ecause of the prop ert y p O δ q δ P α is assumed to satisfy . If K  α ha s o r der type  α , then, by r α, α s -incompactness, p O δ q δ P K is not a co v er of X p β q . Hence , p Z p ε qq ε P g  1 p K q is not a co v er of X p β q . By Lemma 5.3, g  1 p K q has order type  β . Th us, the coun terimage by g of a subset of α of o r der type  α has order t yp e  β . Since W has order t yp e β , then, b y comp osing g with an isomorphism b et w een W and β , we get a function f satisfying the required prop ert y . Not ice that g (hence also f ) is inj ective , since Z p ε q  Z p ε 1 q , for ε  ε 1 . (b) ñ (c) is a particular case of Prop osition 2.5. (c) ñ (a). If (c) holds, then X p β q is not r α, α s -compact, since, b y Lemma 5.3, it is not r β , β s -compact.  R emark 5.5 . Th us, f o r example, for ev ery pair ν ¤ κ of inﬁnite regu- lar cardinals, r κ  κ, κ  κ s -compactness does not imply r κ  ν, κ  ν s - compactness, since there is no function f : κ  ν Ñ κ  κ satisfying Condition (b) in Theorem 5.4. Similarly , r κ 2  κ, κ 2  κ s -compactness do es no t imply r κ  ν, κ  ν s - compactness. ORDINA L COMP ACT NESS 35 Th us, Coro lla ry 2.6(2)(3) cannot b e improv ed. Notice that, b ecause of Theorem 4.5(2) ñ (1), if X is r κ  κ, κ  κ s -compact and not r κ 2 , κ 2 s - compact, then | X | ¡ κ . Corollary 5.6. Supp o s e that α and β ar e nonzer o or dinals, and | α |  | β | . T hen the fo l low ing statements hold. (1) X p β q is r α, α s -c om p act. (2) Ther e is so m e p X , τ q which is r β , β s -c om p act and not r α, α s - c om p act. Pr o of. If f : β Ñ α is an injectiv e function, t hen | α | ¡ | β | , since | α |  | β | . Hence K  f p β q  α has o r der t yp e  α , but f  1 p K q  β has order type β . Hence Condition (b) in Theorem 5.4 fa ils, hence also the equiv alen t Conditions (a) and (c) fail.  Of course, Corollary 5.6 do es not hold in the case when τ is requested to b e closed under unions. See, e. g., Corollary 2.6(6)-(8 ) . The next Theorem is the analogue of Theorem 5.4 in the case when τ is ask ed to b e closed under unions. Theorem 5.7. L et α , β b e nonzer o o r dinals, and assume the notations in Deﬁnition 5.1. Then the fol lowing c on d i tions ar e e quivalent. (a) X U p β q is not r α, α s -c om p act. (b) Ther e e x i s ts a function f : β Ñ α such that, for every K  α with or der typ e  α , it h app ens that f  1 p K q has or der typ e  β . (c) F or every X and τ , if τ is close d under unions, then r α, α s - c om p actness of p X , τ q implies r β , β s -c om p actness of p X , τ q . Pr o of. (a) ñ (b). Supp ose that (a) holds, and that p O δ q δ P α is a coun- terexample to the r α, α s -compactness of X p β q . By the deﬁnition o f τ U , eac h O δ has the form  ε P W δ Z p ε q , for some W δ  β . F or δ P α , let W  δ  W δ z  γ  δ W γ , and let O  δ   ε P W  δ Z p ε q . Notice that p O  δ q δ P α is still a co v er of X p β q , hence it is still a coun terexample to the r α, α s -compactness of X p β q , since O  δ  O δ , for ev ery δ P α . Since p O  δ q δ P α co v ers X p β q , we ha v e  t Z p ε q | ε P W  δ , for some δ P α u   t Z p ε q | ε P  δ P α W  δ u  X p β q , hence, by Lemma 5.3, the order t yp e o f W   δ P α W  δ   δ P α W δ equals β . Let g : W Ñ α b e deﬁned b y g p ε q  the unique δ P α suc h that ε P W  δ . If K  α has order type  α , then, by r α, α s -incompactness, p O  δ q δ P K is not a cov er of X p β q . Hence p Z p ε qq ε P g  1 p K q is not a cov er of X p β q . By Lemma 5.3, g  1 p K q has order type  β . W e ha v e prov ed that the coun terimage by g of a subset of α of order t yp e  α has order t ype  β , th us, arguing as in corresp onding part 36 P AOLO LI PP AR INI of the pro of of Theorem 5.4, and since W has order t yp e β , we get a function f as desired. (b) ñ (c) follows from the la st statement in Prop osition 2.5. (c) ñ (a). If (c) holds, then X U p β q is not r α, α s -compact, since, b y Lemma 5.3, it is not r β , β s -compact, and since τ U is closed under unions.  6. r α, β s -comp a ctness of T 1 sp aces The counterexample s presen ted in Examples 3.2 (2) and 3.10 satisfy v ery few separation axioms. In fact, w e can show that more results ab out r β , α s -compactness can b e pro v ed just on the assu mption that w e are dealing with T 1 top ological spaces. Indeed, since in this note w e ha v e k ept the greatest p ossible generalit y , w e men tion that w e do not actually need a T 1 top ological space, in order to prov e the results in the presen t section. The follow ing we ak er notion is enough. Deﬁnition 6.1. If X is a nonempt y set, and τ is a nonempt y family of subsets of X , we say that p X , τ q is T 1 if and only if, for ev ery O P τ , and ev ery x P O , O zt x u P τ . Clearly , the ab o v e condition is equiv alen t to asking that, for ev ery O P τ , and every ﬁnite F  X , O z F P τ . T rivially , if τ is a top ology on X , then p X , τ q is T 1 in the ab ov e sense if a nd only if it is T 1 in the ordinary top ological theoretical sense. It is con v enient to in tro duce some notation, in order to state the next Prop osition more concisely . Deﬁnition 6.2. If β is an inﬁnite or dina l, w e let β ℓ b e the la rgest limit ordinal ¤ β . Th us, β ℓ  β  n , for an appropriate n P ω . Prop osition 6.3. Supp ose that X is T 1 , a n d let α b e an inﬁnite or d i - nal. (1) X is r α, α s -c om p act if and only if X is r α  1 , α  1 s -c om p act. (2) F or every n P ω and inﬁnite β ¤ α , X is r β , α s -c om p act if and only if it is r β ℓ , α  n s -c om p act. (3) F or every inﬁnite β ¤ α , X is r β , α s -c om p act i f and only if it is r β ℓ , α  ω q -c om p act. (4) If β ¤ α an d β is inﬁnite, then X is r β , α s -c om p act if and only if it is r γ , γ s -c om p act, for every limit or dinal γ with β ℓ ¤ γ ¤ α . Pr o of. (1) One implication follows from Corolla ry 2.6(1) and Prop osi- tion 2.3(1). On the ot her hand, supp ose that X is r α  1 , α  1 s -compact a nd let p O δ q δ P α b e a cov er o f X . Without loss of generalit y , e. g., b y Lemma ORDINA L COMP ACT NESS 37 2.9, we can suppose tha t O 0  H . Let x P O 0 , a nd, for δ P α with δ ¡ 0, let O 1 δ  O δ zt x u . Since p X , τ q is a ssu med to b e T 1 , eac h O 1 δ still b elongs to τ . Moreov er, p O 1 δ q δ P α is still a cov er of X . Notice that every sub co v er o f p O 1 δ q δ P α m ust con tain O 0 , which is the only elemen t of the co v er con taining x . Rearrange p O 1 δ q δ P α as p U δ q δ P α  1 b y letting U δ  O 1 f p δ q , where f : α  1 Ñ α is the bijection deﬁned by f p δ q  $ ' & ' % δ  1 if δ  ω , δ if ω ¤ δ  α, 0 if δ  α. By a pplying r α  1 , α  1 s -compactness to p U δ q δ P α  1 , w e get H  α  1 suc h that H ha s order t ype  α  1, and p U δ q δ P H is a cov er. Since U α  O 1 0 , and O 1 0 is the only elemen t of the cov er con taining x , we hav e that U α b elongs to the subcov er, that is, α P H . Since H has order t yp e  α  1, then necessarily H X α has order t ype  α . Since f | α is order-preserving, t hen a lso f  1 p H X α q has order ty p e  α . Hence K  f  1 p H q , to o, has order type  α , since α is inﬁnite, a nd w e are adding to f  1 p H X α q just one elemen t “a t the b eginning”. Then p O 1 δ q δ P K is a co ver o f X indexed by a set of order type  α , and also p O δ q δ P K is a co v er, since O 1 δ  O δ , f o r ev ery δ P α . Hence, p O δ q δ P K is a sub co v er o f o rder t yp e  α of our original cov er p O δ q δ P α , and w e ha v e prov ed r α, α s -compactness. (2) - (4) are immediate from (1) and Prop osition 2.3.  Of course, Item 1 in Prop osition 6.3 is false without the a ssu mption that α is inﬁnite. Indeed, the discrete space with exactly n elemen ts is r n  1 , n  1 s -compact, but not r n, n s -compact. The next Lemma captures a v ery useful consequen ce of b eing T 1 . Lemma 6 .4. Supp os e that α is an or dinal, cf α  ω , an d p α n q n P ω is a strictly incr e asing se quenc e such that sup n P ω α n  α . If X is T 1 and not r α, α s -c om p act, then ther e is a c ounter e x ample p O δ q δ P α to the r α, α s -c om p actness of X with the p r op erty that, for every n P ω , O α n is indisp ensable (De ﬁ nition 4.1). Pr o of. Let α and the α n ’s b e giv en. Suppo se that p O δ q δ P α is a coun- terexample to r α, α s -compactness. By Lemma 2.9, w e can also supp ose that, for ev ery δ  α , O δ is not con tained in  ε  δ O ε . In particular, for ev ery n P ω , w e can choose x n P O α n suc h that x n R  ε  α n O ε . Deﬁne 38 P AOLO LI PP AR INI p O 1 δ q δ P α as follows . O 1 δ  # O δ if δ ¤ α 0 , O δ zt x 0 , . . . , x n u if α n  δ ¤ α n  1 . Since X is T 1 , eac h O 1 δ still b elongs to τ . Moreo v er, p O 1 δ q δ P α is still a co v er of X . Indeed, f or ev ery n P ω , x n P O 1 α n . If x is not one of the x n ’s, then x P O δ , for some δ P α , a nd also x P O 1 δ . Since O 1 δ  O δ , fo r ev ery δ P α , w e hav e that p O 1 δ q δ P α , to o, is a coun terexample to r α, α s - compactness, and it is easy to see that p O 1 α n q n P ω is a set o f indisp ensable elemen ts. Th us, p O 1 δ q δ P α is a co v er as wan ted.  Man y results on T 1 spaces will b e obtained b e rearranging the indis- p ensable elemen ts giv en by Lemma 6.4. The follo wing notatio n shall b e useful in the pro o f of the forthcoming Theorem 6.6. Deﬁnition 6.5. If β is an y ordinal, let β  b e the smallest o rdinal ¤ β suc h that |r β  , β s| ¤ ω . Th us, β  is the largest ordinal ¤ β whic h is either 0 , or has uncountable coﬁnality , or has coﬁnalit y ω but can b e written as a limit o f ordina ls of uncoun table coﬁnality . Theorem 6.6. Supp ose that X is T 1 , and β is an or dinal of c oﬁnality ω . Then the fol lowing c ondi tion s ar e e quivalent. (1) X is r β , β s -c om p act. (2) X is r β  α, β  α s -c om p act, for every or dina l α with | α | ¤ ω . (3) X is r β  α, β  α s -c om p act, for some or dinal α with | α | ¤ ω . (4) X is r β , β  ω 1 q -c om p act. Pr o of. (2)  (4) follo ws from Prop osition 2.3 (4), hence it is enough to pro v e the equiv alence of (1) - (3). W e shall ﬁrst pro v e the theorem in some particular cases. Claim 1. Conditions ( 1 ) - (3) ar e e quivalent in c ase β  β   ω . Pr o of of Claim 1. In case β   0, (1) ñ (2) follo ws f rom Prop osition 2.3(1) and Corollary 2.6(4) with β  γ  ω . In case β  ¡ 0, (1) ñ (2) follo ws from Prop osition 2.3(4) and Corol- lary 2.6(5), b y ta king t here α  β  , λ  ω and β  β   ω . (2) ñ (3) is tr ivial. W e shall prov e (3) ñ (1) by proving the con trap ositiv e form. So supp ose that X is not r β , β s -compact, and α  ω 1 . W e w an t to show that X is not r β  α, β  α s -compact. F or n  ω , let α n  β   n . Since β  β   ω , then, b y Lemma 6.4, there is some co v er p O δ q δ P β witnessing r β , β s -incompactness, and such that eac h O α n is ORDINA L COMP ACT NESS 39 indisp ensable . If β   0, then r β  α, β  α s -incompactness fo llo ws from Prop osition 4.2(1), hence in what follows let us supp ose β  ¡ 0. F or ev ery H  β  β   ω , if p O δ q δ P H is a co v er of X , then the order t yp e of H is β  β   ω , hence the o rder type of H X β  is β  , since β  is a limit ordinal. Moreov er, H X r β  , β q  r β  , β q , since O δ is indisp ensable , for ev ery δ P r β  , β q . Let f : β   ω  α Ñ β   ω b e a bijection whic h is the iden tit y on β  , and let p U ε q ε P β   ω  α b e deﬁned by U ε  O f p ε q . W e claim tha t p U ε q ε P β   ω  α witnesses that X is not r β   ω  α, β   ω  α s -compact, and this is what w e w an t, since β   ω  α  β  α . Indeed, if K  β   ω  α , and p U ε q ε P K is a cov er of X , then p O δ q δ P H , with H  f p K q , is a co v er of X . Since f is the iden tit y on β  , then, b y the ab o v e men tioned prop erties of H , w e get that the order t ype of K X β  equals the order type o f H X β  , that is, β  ; moreov er, K X r β  , β   ω  α q  r β  , β   ω  α q , thus K has o r der ty p e β   ω  α , hence r β   ω  α, β   ω  α s -incompactness is prov ed.  C l aim 1 Claim 2. Conditions ( 1 ) - (3) ar e e quivalent in the c ase when β  has c oﬁn ality ω , and β  β  . Pr o of of Claim 2. In view of Claim 1, and of Prop osition 6 .3(1), it is enough to sho w that if cf β   ω , then r β  , β  s -compactness is equiv- alen t to r β   ω , β   ω s -compactness. The fo r mer implies the latter b ecause of Corollary 2.6(3) (ta king β  α  β  there), by Prop osition 2.3(4), a nd since w e ha v e a ssum ed that cf β   ω . W e shall pro v e t he rev erse implication by contrapo sition. Supp ose that X is not r β  , β  s - compact. W e w an t to sho w that X is not r β   ω , β   ω s -compact, Cho ose some strictly increasing sequence p α n q n P ω coﬁnal in β  . This is p ossible, since cf β   ω . By Lemma 6.4, there is a coun terexample p O δ q δ P β  to r β  β  s -compactness suc h that eac h O α n is indisp ensable. Th us, if H  β  and p O δ q δ P β  is a cov er of X , then H has order t yp e β  , and moreov er α n P H , for ev ery n P ω . Let A  p β   ω qzt α n | n P ω u . A has order t yp e β   ω , since β  is expressible a s a limit of ordinals of uncountable coﬁnality , hence taking oﬀ a sequence of order t yp e ω do es not alter the order t yp e of β  . Let p O 1 δ q δ P A b e deﬁned by O 1 δ  O δ , if δ P β  zt α n | n P ω u , and b y O 1 β   n  O α n , for n P ω . Since these latt er elemen ts of the co v er are indisp ensable, it is easy t o see that p O 1 δ q δ P A is a coun terexample to r β   ω , β   ω s -compactness.  C l aim 2 Pr o of of T he or em 6.6 (c ontinue d). Summing up, we ha v e prov ed the theorem in the case when either (1) β  β   ω , or 40 P AOLO LI PP AR INI (2) β  β  and cf β   ω . No w let β b e arbitra ry . By deﬁnition, β ¥ β  , and, since w e ha v e assumed cf β  ω , w e ha v e further that, if cf β  ¡ ω , then β ¥ β   ω . Notice also that, by deﬁnition, there is γ with | γ | ¤ ω suc h that β  β   γ and, if cf β  ¡ ω , then, by ab ov e, there is γ 1 with | γ 1 | ¤ ω suc h that β  β   ω  γ 1 . No w observ e that, if the statemen t of the theorem holds for some giv en ordinal β 1 in place of β , and β 2 is another ordinal such that β 2  β 1  γ , for some γ with | γ | ¤ ω , then the statemen t o f the theorem holds for β 2 in place of β , to o . The ab o v e observ ations sho w that the t w o already pro v ed pa r t icu- lar cases (1) a nd (2) imply t he statemen t of the theorem in its full generalit y .  R emark 6.7 . (a) The assumption that β has coﬁnalit y ω in Theorem 6.6 is necessary . By Example 3.2(3), if κ is regular and uncoun table, then p κ, ord q is r κ  ω , κ  ω s -compact, but not r κ, κ s -compact, hence the implication (3) ñ (1) in the statemen t of Theorem 6.6 fails, f or β  κ and α  ω . (b) On the other hand, for β ¥ ω , and T 1 spaces, the implication (1) ñ (2) in Theorem 6.6 a lw ays holds, eve n without the assumption that β has coﬁnality ω . Indeed, b y Prop osition 6 .3(4), r β , β s -compactness implies r β ℓ , β ℓ s -compactness, th us, without loss of generalit y , w e can supp ose that β is limit. Then, for ev ery α  ω 1 , we get r β  α, β  α s - compactness: this f o llo ws from Theorem 6.6 itself, in case cf β  ω , and from Corollary 2.6( 3 ) and Prop osition 2.3(1), if cf β ¡ ω . (c) On the con trary , the implication (1) ñ (2) in the statemen t of Theorem 6.6 fails, in general, for non T 1 spaces. See, for example, the ﬁrst example in Remark 5.5, with κ  ω . (d) Also the implication (3 ) ñ (1) in the statemen t of Theorem 6.6 fails, in general, for non T 1 spaces. Just consider Example 3.2(2), and tak e β  κ  ω and arbitrary α ¡ 1. Corollary 6.8. Supp ose that X is T 1 . Then X is r ω , ω s -c om p act if and only if X is r α, α s -c om p act, for some (e quivalently, every) c ountably inﬁnite or din al α , if and only if X is r ω , ω 1 q -c om p act. Pr o of. The corollary fo llows by taking β  ω in Theorem 6.6.  Theorem 6.6 can b e used to strengthen Prop osition 6.3. ORDINA L COMP ACT NESS 41 Deﬁnition 6.9. Recall from Deﬁnition 6.5 the deﬁnition of β  . F or an ordinal β , deﬁne β  as follows : β   # β  if either cf β   ω , or β  β   n , for some n  ω , β   ω otherwise . Notice that β  ¤ β , for ev ery o r dinal β . Corollary 6.10. S upp ose that X is T 1 , and β ¤ α ar e in ﬁnite or dinals. Then the fol lowing c o n ditions ar e e quivalent. (1) X is r β , α s -c om p act. (2) X is r β  , α  ω 1 q -c om p act. (3) X is b oth r β  , β  s -c om p act, a n d r γ , γ s -c om p act, f o r every γ such that β ¤ γ ¤ α and γ  γ  . Pr o of. (1) ñ (3) F rom Prop osition 2 .3(1) w e get r β , β s -compactness. If cf β   ω , then r β  , β  s -compactness f o llo ws from Theorem 6.6(3) ñ (1), with β  in place of β , and since, b y the deﬁnitions of β  and of β  , w e hav e that β  β   α 1 , f or some α 1 with | α 1 | ¤ ω . If cf β   ω , then β  β   n , fo r some n  ω , and r β  , β  s -compactness fo llo ws from Prop osition 6.3(1), since β is assumed to b e inﬁnite. Finally , r γ , γ s -compactness, for ev ery γ suc h that β ¤ γ ¤ α , is trivial, b y Prop osition 2.3(1). In order to prov e (3) ñ (2), in view o f Prop osition 2.3(4), it is enough to prov e r ε, ε s -compactness, for ev ery ε suc h that β  ¤ ε  α  ω 1 . Let us ﬁx some ε as ab ov e, and let γ  ε  . Notice that γ  γ  , and that γ ¤ α , since |r α, ε s| ¤ ω . If γ ¥ β , then, by assumption, we hav e r γ , γ s -compactness, whic h implies r ε, ε s -compactness, by Theorem 6.6 and Corollary 2.6(3), as remark ed in R emark 6.7(b). O n the other hand, if γ  β , then ε   β  , since β  ¤ β  ¤ ε , and ε   γ  β . Then r β  , β  s -compactness implies r ε, ε s -compactness, aga in by Remark 6.7(b). (2) ñ (1) follows fro m Prop osition 2 .3 (1), since β  ¤ β .  In particular, the compactness prop erties of T 1 spaces are completely determined b y che c king r β , β s -compactness for (1) β ﬁnite, (2) β  ω , (3) β of uncoun table coﬁnalit y (4) β  γ  ω , for γ of uncoun table coﬁnality , a nd (5) β of coﬁnality ω , but expressible as a limit of ordinals of un- coun table coﬁnality . 42 P AOLO LI PP AR INI The ab o v e statemen t, and the next corollar y as w ell, follo w from Corollary 6.10 (1) ñ (3) and the fact that, for inﬁnite β , b oth β  and β  ha v e necessarily one a mong the forms (2)-(4 ). Corollary 6.11. If X is T 1 , and β is the Lindel¨ of or din al o f X , then β has on e of the a b ove forms (1)-(5). In p articular, if β  ω 1 , then β ¤ ω . R emark 6.12 . It f ollo ws from Example 3.2 (3) that the b ehavior o f coun table ordinals in Theorem 6.6 and Corollary 6.8 constitutes an exceptional case. The situation is radically diﬀeren t for lar g er cardi- nals and ordinals, ev en fo r normal to p olo g ical spaces. Indeed, if κ is a regular a nd uncoun table cardinal, then p κ, ord q is r κ  κ, κ  κ s -com- pact but not r κ, κ s -compact. Th us, 6.6 and 6.8 do not hold when ω is replaced b y an uncoun table car dina l. As another example, the disjoint union of t w o copies of p κ, ord q is r κ  κ  κ, κ  κ  κ s -compact, but not r κ  κ, κ  κ s -compact (see Example 3.11). Ho w ev er, Theorem 6.6 do es admit a generalization to larger cardi- nals, but only under a somewhat stronger assumption. Deﬁnition 6.13. If λ is a n inﬁnite cardinal, we sa y that p X , τ q is λ - T 1 if and only if, for ev ery O P τ , a nd ev ery Z  X with | Z |  λ , O z Z P τ . Th us, T 1 is the same as ω - T 1 . If p X , τ q is a T 1 top ological space, and the in tersection of  λ o pen sets of X is still an op en set of X , then p X , τ q is λ - T 1 in the ab ov e sense. Prop osition 6.14. Supp ose that X is λ - T 1 , and β is a limit or dinal of c oﬁna lity ¤ λ . Then the fol lowing c ond itions ar e e quivalent. (1) X is r β , β s -c om p act. (2) X is r β  α, β  α s -c om p act, for every or dina l α with | α | ¤ λ . (3) X is r β  α, β  α s -c om p act, for some or dinal α with | α | ¤ λ . (4) X is r β , β  λ  q -c om p act. The next lemma is prov ed as Lemma 6.4. Lemma 6.15. Supp ose that λ is an inﬁnite c ar dinal, α and γ ar e limit or dinals, γ ¤ λ , cf γ  cf α , and p α ζ q ζ P γ is a strictly i n cr e asing se quenc e such that sup ζ P γ α ζ  α . If X is λ - T 1 and not r α, α s -c om p act, then ther e is a c ounter example p O δ q δ P α to the r α, α s -c om p actness of X with the p r op erty that, for every ζ P γ , O ζ is indisp ensable. ORDINA L COMP ACT NESS 43 Pr o of of Pr op osition 6.1 4. If β is an y ordinal, let β  λ b e the smallest ordinal ¤ β suc h tha t |r β  λ , β s| ¤ λ . Th us, β  λ is t he largest ordinal ¤ β whic h is either 0, or has coﬁnalit y ¡ λ , or can b e written as a limit of ordinals of coﬁnality ¡ λ . The pro of now follo ws t he lines of the pro o f of Theorem 6.6: prov e ﬁrst the result in the case when β  β  λ  λ , a nd then when β  β  λ and ω ¤ cf β  λ ¤ λ .  7. Rela ted notions and problems The spaces in tro duced in Examples 3.2(3) and 3.11 are normal top o- logical spaces (with a base of clop en sets), and they th us pro vide cer- tain limits to prov able results for r β , α s -compactness of normal spaces. Ho w ev er, the theory dev elop ed so far app ears to b e not sharp enough to deal with suc h spaces. As a ve ry roug h hy p othesis, w e conjecture that there is not v ery m uc h diﬀerence in the theory of r β , α s -compactness for, sa y , T 1 spaces and Tyc honoﬀ spaces. W e also conjecture that w e can get some more theorems under the additiona l a ssumption o f normality . All t he ab o v e rough h yp otheses need to be v eriﬁed; the presen t note a pp ears to b e already long enough, thus w e p ostp one the discussion of suc h matters to a subsequen t w ork. Problem 7.1. Give c haracterizations, similar to the ones g iv en in The- orems 5.4 and 5.7, for those pa ir s o f ordinals α and β suc h that r α, α s - compactness implies r β , β s -compactness, for general top ological spaces and, resp ectiv ely , for top ological space s satisfying some giv en separa- tion axiom. O f course, the spaces introduced in Examples 3.2, 3.10, 3.11, 3.12, as we ll as the spaces X τ p β , α q and X T p β , α q of Deﬁnitions 5.1 will b e relev ant to the solution of this problem. R emark 7.2 . Indeed, f o r normal spaces, some problems might b e op en ev en restricted to cardinal compactness. F or example, it is easy to see that X is a linearly Lindel¨ of not Lindel¨ of space (see [AB]) if a nd only if X is r κ, κ s -compact, for ev ery regular uncountable cardinal κ , but there is some uncountable cardinal λ (necessarily o f coﬁnalit y ω ) suc h that X is not r λ, λ s -compact. Problem 7.3. Study the b eha vior of r β , α s -compactness of top olog ical spaces with resp ect to pro ducts. This problem migh t ha v e some interes t, since nontrivial results a bo ut cardinal compactness of pro ducts of top olo g ical spaces are already kno wn. See, e. g ., [Sto, GS, SS, Cai1, Cai2]. See [L i1, Li2] for fur - ther results and references. 44 P AOLO LI PP AR INI Problem 7.4. Study the m utual relationships among r β , α s -compact- ness and other compactness prop erties, either deﬁned in terms of cov- ering prop erties or not. Deﬁnition 7.5. W e can also generalize the presen t notion of ordinal compactness to the relativized notion introduced in [Li3]. If X is a top ological space, a nd F is a family of subsets of X , let us sa y that X is F - r β , α s - c omp act if and only if the follo wing condition holds. F or ev ery sequence p C δ q δ P α of closed sets o f X , if, for ev ery H  α with order type  β , there exists F P F suc h that  δ P H C α  F , then  δ P α C α  H . With this notation, r β , α s -compactness turns out to b e the particular case of F - r β , α s -compactness when F is the set of all singletons of X . The particular case when F is the set of all nonempty op en sets of X migh t hav e particular intere st. The corresp onding not io n when b oth α and β are cardinals has b een studied in [Li4]. Still another g eneralization is suggested by [Li3]. If F is a family o f subsets of X , let us say tha t X is r β , α s - c omp act r elative to F if and only if the following condition holds. F or ev ery sequence p F δ q δ P α of elemen ts of F , if, for ev ery H  α of order t yp e  β ,  δ P H F δ  H , then  δ P α F α  H . F or a top ological space X , r β , α s -compactness is the same as r β , α s -compactness relativ e to the family of a ll closed subsets of X . Problem 7.6. A similar deﬁnition of ordinal compactness can b e give n for abstract logics. See [E] for deﬁnitions and background ab out logics. Let us sa y that a logic L is p α, β q - c omp act if and only if, for ev ery α -indexed set p σ δ q δ P α of L - sen tences, if, f o r ev ery H  α with order t yp e  β , t σ δ | δ P H u has a mo del, then t σ δ | δ P α u has a mo del. Notice the rev ersed order of α and β , to b e consisten t with the stan- dard notation used in the literature ab out compactness of logics. W e do not know whether ordina l compactness fo r log ics is really a new notion, that is, whether or not it can b e expressed in terms of cardinal compactness only . See, e. g., [Ma] for notions of cardinal compactness for logics. The idea o f deﬁning r β , α s -compactness came to us after reading the deﬁnition of an p α, κ q -regular ultraﬁlter in [BK, p. 237]. Deﬁnition 7.7. W e can deﬁne an ev en more general not io n of com- pactness. If Z is any set, and W is a subset of the p o w er set of Z , sa y that a to po lo gical space is r W , Z s -c om p act if and only if, whenev er ORDINA L COMP ACT NESS 45 p O z q z P Z is an op en cov er of X , then there is w P W suc h that p O z q z P w is still a cov er of X . The usual notion o f r µ, λ s -compactness is the particular case when Z has cardinalit y λ , and W is the set of all subsets o f Z of cardinality  µ . More g enerally , our notion of r β , α s -compactness is the particular case when Z  α , and W is the set of a ll subsets of α of or der t yp e  β . W e do not kno w whether there ar e other signiﬁcant pa rticular cases. Reference s [A U] P . Alexandro ﬀ, P . Urysohn, M´ emorie sur les ´ esp ac es top olo giques c omp acts , V er . Ak ad. W e tens ch. Amsterdam 14 (192 9), 1–96 . [AB] A. V. Arhangel’skii, R. Z. Buzyako v a, On line arly Lindel¨ of and str ongly dis- cr etely Lindel¨ of sp ac es , P roc . Amer. Math. So c. 127 (1999), 2449 –2458. [BN] A. Bella, P . Nyikos, Se quential c omp actness vs. c ountable c omp actness , Col- lo q. Math. 120 (2010), 16 5–189. [BK] M. Benda and J. K etonen, R e gularity of ultr aﬁlters , I s rael J. Ma th. 17 (19 7 4), 231–2 40. [Cai1] X. Caicedo, On pr o ductive r κ, λ s -c omp actness, or the Abstr act Comp actness The or em r evisite d , manuscript (199 5). [Cai2] X. Caicedo, The Abstr act Comp actness The or em R evisite d , in L o gic and F oundations of Mathematics (A. Cantini et al. e ditors), Kluw er Academic Publishers (199 9), 131– 141. [Car] P . W. Carruth, Arithmetic of or dinals with applic ations to the the ory of or- der e d Ab elian gr oups , Bull. Amer. Math. 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V a ughan, Count ably c omp act and se quential ly c omp act sp ac es , in Hand- b o ok of set-the or etic top olo gy , edited by K. K unen and J. E. V aug han, North- Holland, Amsterdam, 19 84, Chap. 12, 56 9–602. [V4] J. E. V aughan, Countable Comp actness , in Encyclop e dia of gener al top olo gy , edited by K. P . Hart, J. Nagata and J. E. V aug ha n, Elsevier Science Publishers, Amsterdam, 20 0 4, Chap. d-6, 174 –176. Dip ar timentum Topologiæ (Stic...), Vial e della Ricerca Scientifica, I I Universit ` a di Roma (Tor V er ga t a), I-00133 ROME IT AL Y URL : http: //www.ma t.uniroma2.it/~lipparin

Ordinal Compactness

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