math.GN 2023-11-21 0

Tychonoff Expansions with Prescribed Resolvability Properties

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) and to Comfort and Garc\'ia-Ferreira (2001): (1) Is every $\omega…

Authors: W.W. Comfort, Wanjun Hu

T yc honoff Expansions with Prescri b ed Resolv abilit y Prop erties W.W. Comfo rt ∗ , W anjun Hu † June 4, 2018 Abstract The recen t literature offers examples, s p ecific and hand-crafted, of Tyc honoff spaces ( in ZF C) whic h r esp ond n egativ ely to these questions, due resp ectiv ely to Ceder and Pearson (1967) and to Comfort and Garc ´ ıa-F err eira (2001 ): (1) Is e v- ery ω -resolv ab le space maximally resolv able? (2) Is ev ery maximally resolv able space extraresolv able? No w using the metho d of KI D expansion, the authors sho w that every suitably restricted Tyc honoff top ological space ( X, T ) admits a larger T ychonoff to p olog y (that is, an“expansion”) witnessing su c h failure. Sp ecifically the authors s h o w in ZF C that if ( X, T ) is a m aximally resolv able T yc honoff space with S ( X, T ) ≤ ∆ ( X, T ) = κ , then ( X , T ) has Tyc honoff expansions U = U i (1 ≤ i ≤ 5), w ith ∆( X, U i ) = ∆( X , T ) and S ( X, U i ) ≤ ∆( X , U i ), such that ( X , U i ) is: ( i = 1) ω -resolv able but not maximally resolv able; ( i = 2) [if κ ′ is regular, with S ( X, T ) ≤ κ ′ ≤ κ ] τ -resolv able for all τ < κ ′ , but not κ ′ -resolv able; ( i = 3) maxi- mally reso lv able, but not extraresolv able; ( i = 4) extraresolv able, b ut not maximally resolv able; ( i = 5) maximally resolv able and extraresolv able, but not strongly ex- traresolv able. Keyw ord: R esolv able space, extraresolv able space, strongly e xtraresolv able space, maximally resolv able space, ω -resolv able space, Souslin num b er, indep endent family MSC Primary 05A18, 03E05 , 54 A10; Secondary 03E35 , 54 A25, 05D05 1 In tro duction, Defini t ions and No tation Our principal in terest is in Ty chonoff spaces, i.e., in completely regular, Hausdorff spaces, and all spaces ( X , T ) h yp othesized here, also all expansions (refinemen ts) of T con- structed, will b e T yc honoff topolo gies. The top ological properties w e consider, how ev er, are in telligible (a w onderful w ord in t his context, borrow ed from Hewitt [20]) for a rbitrary spaces, so in 1.2 below, whic h define s the pro p erties w e consider, w e imp ose no separation h yp otheses. ∗ Email: wcomfort@wesley an.edu; Address : Department of Mathema tics and Computer Science, W es- leyan Universit y , W esleyan Station, Middletown, CT 06 459; phone: 860-6 8 5-263 2; F AX 860-68 5-257 1 † Corresp o nding author, Ema il: W anjun.Hu@as urams.edu; Address: Department of Mathematics and Computer Science, Alban y State Univ ersity , Albany , GA 31705; phone: 229- 8 86-47 51 1 Notation 1.1 F or X a set and τ a cardinal, w e set [ X ] τ := { A ⊆ X : | A | = τ } . The sym b ols [ X ] <τ and [ X ] ≤ τ are defined analogously . The sym b ol D ( τ ) denotes t he discre te space of cardina lity τ . When X = ( X, T ) is a space and Y ⊆ X , we denote b y ( Y , T ) the set Y with the subspace top ology inherited from X . The sym b ols w and d denote w eight and density char acter , resp ectiv ely . F or a space X = ( X, T ), the di s p ersion char acter ∆( X ) is the smallest cardinal of an nonempt y op en subset of X , and n wd( X ), the now h er e density numb er of X , is n wd( X ) := min {| A | : A ⊆ X , in t X cl X A 6 = ∅} . Eviden tly n wd ( X ) coincides with the op en density numb er of X [6] defined b y o d( X ) := min { d ( U ) : ∅ 6 = U ∈ T } , whic h has also b een denoted d 0 ( X ) [26]. As in [8] and [25], a subset D of a space X = ( X, T ) is τ - dense in X if | D ∩ U | ≥ τ whenev er ∅ 6 = U ∈ T . It is obv ious that if D is de nse in a space X with n wd ( X ) ≥ τ , then D is τ -dense in X . ( X , T ) is cr owde d if no p oin t of X is isolated in the t o p ology T . (This term, in tro duced b y v an Dou we n [13], has b een a dopted subsequen tly by man y authors [14], [22], [25]. Others hav e called suc h a space dense-in-itself [7].) A family of nonem pty pair wise disjoin t op en subsets of X = ( X , T ) is a c el lular family , and S ( X ), t he Souslin n umb er of X , is S ( X ) := min { κ : no cellular U ⊆ T satisfies |U | = κ } . Definition 1.2 Let X = ( X, T ) b e a space. Then X is (i) r esolvable (Hewitt [2 0]) if it has t wo complemen tary dense subsets; (ii) κ -r esolvable (Ceder [2]) if there is a family of κ -many pa ir wise disjoint dense subs ets of X ; (iii) ma ximal ly r esolvable (Ceder [2 ]) if it is ∆( X )- r esolv able; (iv) extr ar esolva ble (Malykhin [28 ]) if there is a family D of dense subsets, with |D | ≥ (∆( X ) ) + , suc h that ev ery t wo elem ents of D hav e intere section whic h is no where dense in X ; a nd (v) str ongly ex tr ar eso l v able (Comfor t and Garc ´ ıa-F erreira [4], [5]) if there is a fa mily D o f dense subsets, with |D | ≥ (∆( X )) + , suc h that distinct D 0 , D 1 ∈ D satisfy | D 0 ∩ D 1 | < n wd( X ). Remark 1.3 In early v ersions of this manusc ript, circulated priv ately to selected col- leagues, w e w ere able to establish item ( i = 4 ) of the Abstract, ev en it s sp ec ial case Theorem 3.9, only under the additional assumption that there exists a cardinal τ such that τ < κ < 2 τ . Indeed, alt ho ugh w e had s hown in [8] the exis tence of extraresolv able T yc honoff spaces whic h are not maximally resolv able when GCH fails, it w a s a n unsolv ed problem whether suc h spaces exist in ZFC . That question has been settled affirmative ly b y Juh´ asz, Shelah and Soukup [27]. W e are grateful to those authors for furnishing us with a pre-publication cop y of their w ork. 2 Definition 1.4 Let κ ≥ ω . (a) A pa r t ition B of κ is a κ -partition if each B ∈ B satisfies | B | = κ ; (b) a family B = {B t : t ∈ T } of partitions B = {B α t : α < κ t } of κ is τ - indep endent (with 1 ≤ τ ≤ κ ) if | T t ∈ F B f ( t ) t | ≥ τ for each F ∈ [ T ] <ω and f ∈ Π t ∈ F κ t . (c) a family B = {B t : t ∈ T } of indexed partitions B t = { B α t : α < κ t } (with 2 ≤ κ t ≤ κ for eac h t ∈ T ) se p ar ates p oints [resp., sep ar ates smal l sets ] if f o r distinct x, x ′ ∈ κ there are B t ∈ B and (distinct) α, α ′ < κ t suc h that x ∈ B α t and x ′ ∈ B α ′ t [resp., for disjoin t S, S ′ ∈ [ κ ] <κ there are B t ∈ B and (distinct) α, α ′ < κ t suc h that S ⊆ B α t and S ′ ⊆ B α ′ t ]. It is ob vious that an y par t ition in a κ -independen t family (of partitions of κ ) is necessarily a κ - partition. Discussion 1.5 Giv en a p o in t-separating family B as in Definition 1.4, w e denote b y T B the smallest top olog y on κ in whic h eac h set B α t ∈ B t ∈ B is op en; clearly eac h suc h B α t is T B -closed, a nd { T t ∈ F B f ( t ) t : F ∈ [ T ] <ω , f ∈ Π t ∈ F κ t } is a basis for T B . (This is a Hausdorff top olog y since B separates p o in ts of κ , hence is a T yc honoff topolog y since it has a clop en basis.) The ev alua t io n map e B : ( κ, T B ) → Π t ∈ T D ( κ t ) given b y ( e B x ) t = α if x ∈ B α t ( x ∈ κ, t ∈ T , α < κ t ) is a homeomorphism from ( κ, T B ) on to a subspace X o f the T ychonoff space K := Π t ∈ T D ( κ t ). That X := e B [ κ ] is dense in K follows from t he fact that B is 1-indep enden t. Con v ersely , giv en K = Π t ∈ T D ( κ t ) with | T | = 2 κ and with 2 ≤ κ t ≤ κ for eac h t ∈ T , the Hewitt-Marczewski-P o ndiczery theorem (cf. [16](2 .3 .15), [11]( § 3 and Notes)) giv es a dense set X ⊆ K suc h that | X | = κ , and t hen the family B := {B t : t ∈ T } with B t := { π − 1 t ( { α } ∩ X : α < κ t } is a 1-indep enden t family of partitio ns of κ (the set κ here being iden tified with the s ubspace X of K ). One ma y ensure that eac h B t ∈ B is a κ -partition b y t he follo wing device ( here we argue m uc h a s in [7 ](3.8) and [8](1.5)): Give eac h space D ( κ t ) the structure of a top o lo gical group, so that K is a top ological group, let X ∗ b e dense in K with | X ∗ | = κ , and with h X ∗ i the subgroup of K g enerated b y X ∗ let X b e the union of κ - ma ny cosets of h X ∗ i in K . Then B α t := π − 1 t ( { α } ) ∩ X satisfies | B α t | = κ for eac h α < κ t , t ∈ T ; indeed more generally eac h basic op en set U in X (of the form U = ( T n i =1 π − 1 t i ( { α i } )) ∩ X , with α i < κ t i , n < ω ) satisfies | U | = κ , so the family B is ev en κ -independen t, and ∆( X ) = κ . The correspo ndence B ↔ X just described is of Galo is type in the sense that when dense X ⊆ K = Π t ∈ T D ( κ t ) is given with | X | = κ and the family B = {B t : t ∈ T } is defined, then e B : ( κ, T B ) → K satisfie s e B [ κ ] = X . In this pap er in this context, T and { κ t : t ∈ T } b eing giv en, w e use the notations ( κ, T B ), ( X , T B ) and e B [ κ ] inte rchangeably . The p oin t- separating family describ ed in D iscussion 1 .5 may b e chose n to sep- arate small sets in a strong sense . Lemma 1.6, whic h exploits a tric k in tro duced by Ec k ertson [14 ] in a related contex t, strengthens a statemen t giv en in our works [6] a nd [7](3.3(b)). When reference is made, in Lemma 1.6 and la t er, to a pa r t ition { T ( λ ) : λ ∈ Λ } of T , the trivial (one-cell) pa r t it ion is not excluded. 3 Lemma 1.6 L et κ ≥ ω an d | T | = 2 κ , and for t ∈ T let 2 ≤ κ t ≤ κ . L et { T ( λ ) : λ ∈ Λ } b e a p a rtition o f T , with e ach | T ( λ ) | = 2 κ . Then ther e is a κ -indep end e nt family I = {I t : t ∈ T } of p artitions of κ , with |I t | = κ t for e ac h t ∈ T , such that for every or der e d p air ( S, S ′ ) of disjo i n t elements o f [ κ ] <κ and fo r every λ ∈ Λ ther e ar e infinitely ma ny t ∈ T ( λ ) such that S ⊆ I 0 t and S ′ ⊆ I 1 t . Pro of. Let B = {B t : t ∈ T } b e a p oint-separating κ -indep enden t family of partitions of κ with | T | = 2 κ and with |B t | = κ t for eac h t ∈ T , a s giv en in D iscussion 1.5. F or λ ∈ Λ let { T ( λ, ξ ) : ξ < 2 κ } b e a partition of T ( λ ) with eac h | T ( λ, ξ ) | = ω , and using | [ κ ] <κ | ≤ 2 κ let { ( S ξ , S ′ ξ ) : ξ < 2 κ } list all ordered pairs of disjoint mem b ers of [ κ ] <κ (with rep etitions p ermitted). Then define I = {I t : t ∈ T } with I t = { I α t : α < κ t } as follows : if t ∈ T ( λ, ξ ), then I 0 t = ( B 0 t ∪ S ξ ) \ S ′ ξ , I 1 t = ( B 1 t ∪ S ′ ξ ) \ S ξ , and I α t = B α t \ ( S ξ ∪ S ′ ξ ) for 2 ≤ α < κ t . Then eac h I t is a partitio n of κ , a nd since B α t △ I α t ∈ [ κ ] <κ (*) for eac h t ∈ T and α < κ t with B t a κ -partition, so also is eac h I t a κ -pa r t ition. F urther for eac h pair ( S, S ′ ) = ( S ξ , S ′ ξ ) w e ha v e S ⊆ I 0 t and S ′ ⊆ I 1 t for eac h t ∈ T ( λ, ξ ) ∈ [ T ( λ )] ω , as required.  Definition 1.7 With { κ t : t ∈ T } and { T ( λ ) : λ ∈ Λ } given as in Lemma 1.6, a κ -indep enden t fa mily I of partitions of κ with the additional property giv en there is a str ong smal l-set-sep ar ating family of partitions whic h r esp e cts the partition { T ( λ ) : λ ∈ Λ } of T . Remark 1.8 Clearly a κ - indep enden t family {I t : t ∈ T } of partitions of κ , if it resp ects some partit io n { T ( λ ) : λ ∈ Λ } of T , also respects the trivial ( o ne-cell) part it ion. Usually in this pap er w e apply Lemma 1.6 only in the con t ext of the trivial partition; in what follo ws, if no exp licit reference is made to the partition whic h a stro ng small-set-se para ting family of κ - partitions resp ects, w e in tend by default the t rivial partition. The follo wing theorem augmen ts, simplifies and extends argumen ts giv en in our w orks [7](3.8) and [8](1.6 ). As usual w hen a p o in t-separating family I of partitions of κ is giv en, w e do not distinguish notationally b et we en κ and the space X := e I [ κ ] ⊆ K = Π t ∈ T D ( κ t ), nor b et we en a set I α t ∈ I t ∈ I and its image e I [ I α t ] in X . Theorem 1.9 L et κ ≥ ω and | T | = 2 κ , and for t ∈ T let 2 ≤ κ t ≤ κ . Th e n ther e is a κ - indep en dent family I = { I t : t ∈ T } of p artitions of κ with the str ong smal l-set-sep ar ating pr op erty, and with | I t | = κ t for e ach t ∈ T , such that the sp ac e X := e I [ κ ] ⊆ K := Π t ∈ T D ( κ t ) has these pr op erties: (a) X is dense in K ; (b) X is κ -r esolvable; (c) | X | = ∆( X ) = n wd( X ) = κ ; and (d) e ach S ∈ [ X ] <κ is close d and discr ete in X . 4 Pro of. Let T := T ∪ { t } with t / ∈ T , and set κ t := κ . Apply Lemma 1 .6 with { T } the one-cell partition of T : There is a κ -indep enden t family I = {I t : t ∈ T } of κ -partitions of κ with the strong small-set-separating pro p ert y , with |I t | = κ t for eac h t ∈ T (in particular, with |T t | = κ t = κ ). By the argumen t giv en in Discussion 1.5 the set X := e I [ κ ] is dense in K := Π t ∈ T D ( κ t ), so (a) is prov ed. F o r F ∈ [ T ] <ω and f ∈ Π t ∈ F κ t and each I α t (with α < κ t = κ ) w e hav e | ( T t ∈ F I f ( t ) t ) ∩ I α t | = κ (*) since the family I is κ -independen t. Relation (*) sho ws that eac h set e I [ I α t ] is de nse in X (th us pro ving (b)), and it sho ws also that | X | = ∆( X ) = κ . Since X is a cro wded space, ev ery closed, discrete subspace of X is nowhe re dense; so the relation nwd( X ) = κ will follow fro m (d). Giv en S ∈ [ κ ] <κ and x ∈ κ \ S , there is t ∈ T suc h that x ∈ I 0 t and S ⊆ I 1 t ; s ince I 0 t and I 1 t are disjoin t and clopen in X , w e conclude that S is closed. Similarly if x ∈ S ∈ [ κ ] <κ there is t ∈ T suc h that x ∈ I 0 t and S \{ x } ⊆ I 1 t , so I 0 t ∩ S = { x } ; it follo ws that S is dis crete.  Remarks 1.10 (a) In earlier w ork [8] b y a differen t argumen t we ha v e demonstrated the existence of a κ -resolv able dense subset X of some spaces of the form Π t ∈ T D ( κ t ) with | T | = 2 κ , ev en with | X | = ∆( X ) = n wd( X ) = κ . (See also [6](5.3 and 5.4 ) for sim ilar results.) The a rgumen t o f Theorem 1 .9 is preferable, b oth b ecause of its simplicit y and b ecause it giv es in concrete form a family I for wh ich X = e I [ κ ]; this latter feature is essen tial in the proof of Lemma 3.7 below . (b) The case in Definition 1.4 in whic h t here is λ ∈ [2 , κ ] suc h t ha t κ t = λ fo r all t ∈ T , together with passage in that case from B to the space ( κ, T B ) = ( X, T B ), has b een used by man y authors in connection with resolv abilit y questions [13], [6], [7], [25], [8]. 2 The K I D E xp ansion: T ransfer from T to T KI D Here we explain and dev elop furt her the techn iques o r iginating in [21], [22]. In broad terms the goal, giv en a cro wded T yc honoff space ( X , T ), is to augmen t (“expand”) the top ology T to a larger cro wded T yc honoff top ology T KI D in suc h a w ay that certain s p ecified T - dense subsets of X remain T KI D -dense, while certain other subsets of X b ec ome closed and discrete in the top olog y T KI D . In Definition 2.2, the transition fro m T to the T KI D -op en sets W α t is effected via the in t ermediate sets H α t . Their definition dep ends on the h yp ot hesized dense array D and the κ -independen t family I , but not on t he family K . The following notation is as in [7](3.2). Notation 2.1 Let X b e a se t with | X | = κ ≥ ω , and let D = { D γ η : γ < τ , η < κ } b e a partition of X with 1 ≤ τ ≤ κ . Then for S ⊆ κ the set X ( S ) ⊆ X is defined b y X ( S ) := S { D γ η : γ < τ , η ∈ S } . Definition 2.2 Let ( X , T ) b e a cro wded T yc honoff space with | X | = κ ≥ ω , fix nonempt y Z ⊆ X , and let I = {I t : t ∈ Z × 2 κ } b e a p o in t-separating κ -indep endent family of par- titions of κ with I t = { I α t : α < κ t } , 2 ≤ κ t ≤ κ for eac h t ∈ Z × 2 κ . Let 1 ≤ τ ≤ κ and D = { D γ η : γ < τ , η < κ } b e a partition of X , and for t ∈ Z × 2 κ and α < κ t set 5 H α t := X ( I α t ) = S { D γ η : γ < τ , η ∈ I α t } . Let K = { K ξ : ξ < 2 κ } ⊆ P ( Z ), a nd for t = ( x, ξ ) ∈ Z × 2 κ and α < κ t define W α t as follows: If K ξ = ∅ , then W α t = H α t . If K ξ 6 = ∅ , then W 0 t = ( H 0 t ∪ K ξ ) \{ x } , W 1 t = ( H 1 t \ K ξ  ∪ { x } , and W α t = H α t \ ( K ξ ∪ { x } ) fo r 2 ≤ α < κ t . F or each t ∈ Z × 2 κ set H t := { H α t : α < κ t } and W t := { W α t : α < κ t } , and set H := {H t : t ∈ Z × 2 κ } , and W := {W t : t ∈ Z × 2 κ } . Then T I D is the smallest top ology on X suc h that T ⊆ T I D and each H t ⊆ T I D , and T KI D , the KI D exp ansion of T , is the smallest top ology on X suc h that T ⊆ T KI D and each W t ⊆ T KI D . Remarks 2.3 (a) The index ings D = { D γ η : γ < τ , η < κ } and I = {I t : t ∈ Z × 2 κ } in Definition 2.2 are faithful. No suc h restriction is imp osed on the indexing K = { K ξ : ξ < 2 κ } . Indeed in some of the applications w e will ha v e K ξ = ∅ for many ξ < 2 κ . (b) F or t ∈ Z × 2 κ the family H t is a partition of X in to T I D -op en subsets, so eac h H α t is T I D -clop en. Similarly , sinc e for t ∈ Z × 2 κ the f a mily W t is a partition of X in to T KI D -op en sets, also eac h W α t is T KI D -clop en. It then follows, as is required of ev ery top ology hypothesized or constructed in this pap er, that: (c) Each space of the form ( X, T I D ), and eac h space of t he form ( X , T KI D ), is a T yc honoff space. (d) the top ology T KI D dep ends not only on the families K , I , and D , but also on the c hoice of the nonempt y set Z ⊆ X . Our notation do es not reflect that fa ct. No confusion w ith ensu e, indeed in (nearly) all the applications we tak e Z = X . Briefly in Theorem 3.8 we will inv ok e the general theory in the sp ecial case | Z | = 1. T o a v oid irrelev ancies we ga v e Definition 2.2 in uncluttered la nguage, but in fact w e will use the expansion T KI D only when the follo wing additional conditions are sat- isfied. E xcept when noted otherwise, we assu me these henceforth throughout this Sec- tion. F urthermore when families I , D and K ha ve been construc ted or h yp othesized and I α t ∈ I t ∈ I , it is understoo d that the sets H α t and W α t are defined as in Definition 2.2. Standing Hypotheses and Notation 2.4 (1) | X | = ∆( X, T ) = κ ; (2) the indexe d family D is a dense partition of ( X , T ), and D γ := S η<κ D γ η for γ < τ ; (3) the fa mily I = {I t : t ∈ Z × 2 κ } has the strong small-set-separating property; (4) if F ∈ [2 κ ] <ω then S ξ ∈ F K ξ ∈ K ; and (5) ξ < 2 κ , γ < τ ⇒ in t ( D γ , T I D ) ( K ξ ∩ D γ ) = ∅ . 6 Lemma 2.5 [With the con v en tions of 2 .2 and 2.4.] Fix γ < τ and ξ < 2 κ . Then (a) K ξ is close d in ( Z, T KI D ) ; (b) ( K ξ , T KI D ) is d iscr ete; and (c) if ∅ 6 = U ∈ T , H = T t ∈ F H f ( t ) t and W = T t ∈ F W f ( t ) t with F ∈ [ Z × 2 κ ] <ω and f ∈ Π t ∈ F κ t , then | D γ ∩ U ∩ H | = | D γ ∩ U ∩ W | = κ . Pro of. (a) If x ∈ Z \ K ξ then with t := ( x, ξ ) w e hav e x ∈ W 1 t ∈ T KI D and W 1 t ∩ K ξ = ∅ . (b) If x ∈ K ξ then with t := ( x, ξ ) w e ha ve W 1 t ∈ T K I D and W 1 t ∩ K ξ = { x } . (c) Let I := T t ∈ F I f ( t ) t . Since I is κ -independen t w e ha v e | I | = κ . F or eac h η ∈ I the set D γ ∩ H con tains the set D γ η ; sin ce the sets D γ η ( η ∈ I ) are pairwise dis jo int, each dense in ( X, T ), we ha v e κ = | X | ≥ | D γ ∩ U ∩ H | ≥ | I | = κ . (*) It remains to sho w that | D γ ∩ U ∩ W | = κ . F irst, set K := S ( x,ξ ) ∈ F K ξ and L := S ( x,ξ ) ∈ F ( K ξ ∪ { x } ), and note fro m (4) and (5) of 2.4 that in t ( D γ , T I D ) ( D γ ∩ K ) = ∅ , hence a lso in t ( D γ , T I D ) ( D γ ∩ L ) = ∅ (**) (since ( D γ , T I D ) is cro wded). No w let A := ( D γ \ L ) ∩ ( U ∩ H ). Since D γ ∩ U ∩ W ⊇ A , it suffices to show | A | = κ . If A ∈ [ X ] <κ then, writing S := { η < κ : A ∩ D γ η 6 = ∅} , w e hav e | S | ≤ | A | < κ , so b y 2.4(3) there is e t ∈ ( Z × 2 κ ) \ F suc h that S ⊆ I 0 e t ; then S ∩ I 1 e t = ∅ and henc e A ∩ H 1 e t = ∅ . Then with e f := f ∪ { ( e t, 1) } ∈ Π t ∈ F ∪{ e t } κ t and e H := T t ∈ F ∪{ e t } H f ( t ) t = H ∩ H 1 e t ∈ H w e ha ve ∅ = A ∩ H 1 e t = ( D γ \ L ) ∩ ( U ∩ H ) ∩ H 1 e t = ( D γ \ L ) ∩ ( U ∩ e H ) and hence D γ ∩ L ⊇ ( D γ ∩ L ) ∩ ( U ∩ e H ) = ∅ ∪ [( D γ ∩ L ) ∩ ( U ∩ e H )] = [( D γ \ L ) ∩ ( U ∩ e H )] ∪ [( D γ ∩ L ) ∩ ( U ∩ e H )] = D γ ∩ U ∩ e H . (***) By (*) applied with e H replacing H , the set D γ ∩ U ∩ e H is a nonempty T I D -op en subset of D γ , so (***) contradicts (**).  Corollary 2.6 [With the conv en tio ns of 2.2 and 2.4.] (a) ( D γ , T I D ) is c r owde d, and D γ is dense in ( X, T I D ) ; (b) ( D γ , T KI D ) is cr owde d, and D γ is dense in ( X , T KI D ) ; and (c) ∆( X, T I D ) = ∆( X, T K I D ) = ∆( X, T ) = κ . Pro of. The inequalities ∆( X, T I D ) ≤ ∆( X, T ) = κ and ∆( X , T KI D ) ≤ ∆( X, T ) = κ of (c) follo w from the inc lusions T ⊆ T I D and T ⊆ T KI D , and all else is immediate from Lemma 2.5.  It is easily seen that eac h infinite (Hausdorff ) space ( X , T ) con tains an infinite cel- lular fa mily , hence satisfies S ( X , T ) ≥ ω + . According to a result o f Erd˝ os and T arski [17 ] (see also [11](3.5), [12](2.14)) ev ery infinite Souslin n um b er is regular. That allo ws us to compute exac tly num b ers of the form S ( X , T KI D ) in terms of the n umber S ( X , T ) and the family { κ t : t ∈ Z × 2 κ } . 7 Lemma 2.7 [With the con v en tions of 2 .2 and 2.4.] S ( X , T KI D ) is the smal le st r e gular c ar dinal κ ′ such that (i) κ ′ ≥ S ( X, T ) , and (ii) t ∈ Z × 2 κ ⇒ κ ′ ≥ κ + t . Pro of. F rom T ⊆ T KI D follo ws S ( X , T ) ≤ S ( X, T KI D ). F urther for t ∈ Z × 2 κ the family { W α t : α < κ t } is cellular in ( X, T KI D ), so S ( X, T KI D ) ≥ κ + t . Since S ( X , T K I D ) is regular by the cited t heorem of Erd˝ os and T arski, w e ha v e S ( X, T KI D ) ≥ κ ′ . Supp ose now that { U ζ ∩ W ζ : ζ < κ ′ } is a faithfully indexed cellular family of T KI D -basic op en subsets of X ; here U ζ ∈ T and W ζ = T t ∈ F ζ W f ζ ( t ) t with F ζ ∈ [ Z × 2 κ ] <ω , f ζ ∈ Π t ∈ F ζ κ t , W f ζ ( t ) t ∈ W . Since { F ζ : ζ < κ ′ } is a family of finite sets indexed (not necessarily faithfully) by the regular cardinal κ ′ , there are A ∈ [ κ ′ ] κ ′ and a set F such that F ζ 0 ∩ F ζ 1 = F fo r ev ery pair { ζ 0 , ζ 1 } ∈ [ A ] 2 . (See [11] o r [12 ] or [24] f or pro ofs and bibliographic comme ntary on this theorem, its sp ecial cases and generalizations.) Since | F | < ω and f ζ ( t ) < κ t < κ ′ for eac h ζ ∈ A and t ∈ F , there is B ∈ [ A ] κ ′ suc h that f ζ 0 ( t ) = f ζ 1 ( t ) for all ζ 0 , ζ 1 ∈ B and t ∈ F . W e defi ne f : F ζ 0 ∪ F ζ 1 → S t ∈ F ζ 0 ∪ F ζ 1 κ t b y f ( t ) =    f ζ 0 ( t ) = f ζ 1 ( t ) if t ∈ F f ζ 0 ( t ) if t ∈ F ζ 0 \ F f ζ 1 ( t ) if t ∈ F ζ 1 \ F    . (More succin tly: f = f ζ 0 | F ζ 0 ∪ f ζ 1 | F ζ 1 .) Then since S ( X , T ) ≤ κ ′ = | B | there are distinct ζ 0 , ζ 1 (henceforth fixed) in B such that U ζ 0 ∩ U ζ 1 6 = ∅ . Then H ζ 0 ∩ H ζ 1 = T t ∈ F 0 ∪ F 1 H f ( t ) t , and (using (c) in Lemma 2 .5) w e hav e ∅ 6 = ( H ζ 0 ∩ H ζ 1 ) ∩ ( U ζ 0 ∩ U ζ 1 ) ∈ T I D . No w choose and fix γ < τ , and (arguing m uc h as in the pro of of Lemma 2.5( c)) set K := S ( x,ξ ) ∈ F ζ 0 ∪ F ζ 1 K ξ and L := S ( x,ξ ) ∈ F ζ 0 ∪ F ζ 1 ( K ξ ∪ { x } ); then K ∈ K b y 2.4(4) and D γ \ K is dens e in the crow ded s pace ( D γ , T I D ) b y 2.4(5), so D γ \ L is a lso dense in ( D γ , T I D ), hence also in ( X , T I D ) b y Corollary 2.6(a). Then ( D γ \ L ) ∩ ( H ζ 0 ∩ H ζ 1 ) ∩ ( U ζ 0 ∩ U ζ 1 ) 6 = ∅ , so ( D γ \ L ) ∩ ( W ζ 0 ∩ W ζ 1 ) ∩ ( U ζ 0 ∩ U ζ 1 ) 6 = ∅ , con trary to the condition ( W ζ 0 ∩ U ζ 0 ) ∩ ( W ζ 1 ∩ U ζ 1 ) = ∅ .  Discussion 2.8 The metho d of K I D expansion was intro duced in [21] and was used in [22] to giv e the existence, assuming Lusin’s Hyp othesis, of ω -resolv able T yc honoff spaces which are not maximally resolv able. The presen t authors ha ve used the method subseque ntly [7], [8] to find and construct explicit spaces with some of the prop erties giv en in the Abstract. Argumen ts with some s imilar features w ere found indep enden t ly and exploited b y Juh´ asz, Sze ntmik lossy , and So ukup [25]; w e note that [25] w as subm itted to the journal of record b efore [8] w a s submitted, furthermore the date of publication of [25] precedes that of [8]. 8 The principal thrust of the presen t pap er is this: Not only do sp ecific spaces (constructed as in [21], [22], [7], [25], [8]) exist with the prop erties lis ted, but indeed ev ery cro wded T yc honoff space sub ject to minimal necessary conditions admits suc h T yc honoff expansions. Definition 2.9 [With the con ve ntions of 2.2, but with K not y et defined.] Let M = { M ξ : ξ < 2 κ } ⊆ P ( Z ) with M 0 = ∅ . Then f M = { f M ξ : ξ < 2 κ } is defined as follo ws. f M 0 = ∅ , and if 0 < ξ < 2 κ and f M η has b een defined for all η < ξ then f M ξ = M ξ if eac h set of the for m ( M ξ ∪ g M η 0 ∪ g M η 1 ∪ · · · ∪ g M η m ) ∩ D γ ( m < ω , η i < ξ , γ < τ ) has empt y in terior in the space ( D γ , T I D ), f M ξ = ∅ otherwise. Lemma 2.10 L et Y b e a cr owde d (Hausdorff ) sp ac e and let E = S i ≤ m E i ⊆ Y with e ach E i discr ete, m < ω . Then in t Y E = ∅ . Pro of. This is clear when m = 0. Supp ose it holds for m = k and let E = S i ≤ k +1 E i ⊆ Y with eac h E i discrete. Suppose for a contradiction that t here is p ∈ in t Y E , sa y with p ∈ E k +1 , and find op en U ⊆ Y suc h that U ∩ E k +1 = { p } . Then ( U ∩ in t Y E ) ∩ E k +1 = { p } , so S i ≤ k E i con tains the nonem pty op en set ( U ∩ in t Y E ) \{ p } .  Theorem 2.11 [With the con v en tions of 2.2 and 2.4(1), (2), (3).] L et M = { M ξ : ξ < 2 κ } = P ( Z ) and let K := f M = { f M ξ : ξ < 2 κ } . Then (a) K satisfies c onditions (4) and ( 5 ) of 2.4 ; (b) if ξ < 2 κ and in t ( D γ , T KI D ) ( M ξ ∩ D γ ) = ∅ for al l γ < τ , then M ξ = f M ξ ∈ K ; and (c) e ach sp ac e ( D γ ∩ Z , T KI D ) is h er e ditarily irr esolvable. Pro of. (a ) is obv ious. (b) F ix ξ < 2 κ and γ < τ , and let η 0 , η 1 , · · · η m < ξ , ∅ 6 = U ∈ T and H = T t ∈ F H f ( t ) t with F ∈ [ Z × 2 κ ] <ω , f ∈ Π t ∈ F κ t . W e must show that if in t ( D γ , T KI D ) ( M ξ ∩ D γ ) = ∅ for all γ < τ , then in t ( D γ , T I D ) (( M ξ ∪ g M η 0 ∪ g M η 1 ∪ . . . ∪ g M η m ) ∩ D γ ) = ∅ . (*) W riting W = T t ∈ F W f ( t ) t , we ha ve , since D γ \ M ξ is dense in ( D γ , T KI D ) and ∅ 6 = U ∩ W ∈ T KI D , that Y := ( D γ \ M ξ ) ∩ ( U ∩ W ) is dense in (( D γ ∩ ( U ∩ W )) , T KI D ). F urther since ( D γ ∩ ( U ∩ W ) , T KI D ) is cro wded, its dense subset ( Y , T KI D ) is cro wded. W e ha ve W \ H ⊆ L := S ( x,ξ ) ∈ F ( K ξ ∪ { x } ), with L the union o f finitely man y discrete subsets of ( Z, T KI D ) ⊆ ( X, T KI D ). Eac h g M η i ∈ K is also discrete in ( Z, T KI D ) ⊆ ( X , T KI D ), so fro m Lemma 2.10 it follows that the set Y \ ( S i ≤ m g M η i ∪ L ) remains dense in ( D γ ∩ U ∩ W , T KI D ), and (*) follo ws. (c) Supp ose for some γ 0 < τ there are ξ 0 < 2 κ and nonempty S ⊆ D γ 0 ∩ Z suc h that M ξ 0 ⊆ S and b oth M ξ 0 and S \ M ξ 0 are dens e in ( S, T KI D ). F rom in t ( S, T KI D ) M ξ 0 = ∅ 9 it follo ws that int ( D γ 0 , T K I D ) M ξ 0 = ∅ , so int ( D γ , T K I D ) ( M ξ 0 ∩ D γ ) = ∅ for each γ < τ . F ro m (b) we then hav e M ξ 0 = g M ξ 0 ∈ K , so b y Lemm a 2.5(a) the set M ξ 0 is closed in ( Z, T KI D ) (hence in ( S, T KI D )); this contradicts the dens ity in ( S, T KI D ) of b oth M ξ 0 and S \ M ξ 0 .  3 The K I D E xp ansion: Applicatio n s W e b egin this Section b y proving ( t he case | X | = ∆( X ) of ) our principal theorem (cf. item ( i = 1) of the Abstract). The result is in the t r a dition of the sev eral pap ers listed in the Bibliograph y whic h resp ond to the Ceder-Pe arson ques tio n (Is there an ω -resolv able space whic h is not maxim ally resolv able?), but this has a differen t fla v or: Not only can examples of suc h spaces b e cons tructed b y ad h o c tech niques, but indeed every (suitably restricted) ω -resolv able Tyc honoff space a dmits a T yc honoff expansion U suc h that ( X , U ) remains ω -resolv able but is not maxim ally resolv able. F or remarks in tended to justify or to explain the sp ecial h yp othesis “ S ( X , T ) ≤ | X | ” in Theorem 3.1, see Remark 5.3 b elow, where it is noted that in some settings where S ( X, T ) ≤ | X | fails, ω -resolv a bility implies maximal resolv ability . Theorem 3.1 L et X = ( X, T ) b e a cr o w de d, ω -r e solvable T ychonoff sp ac e with S ( X , T ) ≤ | X | = ∆( X , T ) = κ . The n ther e is a T ychonoff r efine m ent U of T such that (a) S ( X , U ) = S ( X, T ) and ∆( X, U ) = ∆( X, T ) ; (b) ( X, U ) is ω -r esolvable; (c) ( X, U ) is not maximal ly r es o lvable; and (d) ( X, U ) is not S ( X, T ) -r esolvable, if ( X, T ) is maximal ly r e solvable. Pro of. If ( X , T ) is not maximally resolv able the conditions are satisfied with U := T , so w e assume in what follows that ( X , T ) is maximally r esolv able. Let D = { D n η : η < κ, n < ω } b e a faithfully indexed dense partitio n of ( X, T ), and set D n := S η<κ D n η for n < ω . T ak e Z = X in Definition 2.2 and let I = {I t : t ∈ X × 2 κ } b e a κ -indep enden t family of partitio ns I t of X with the strong small-set-separating prop ert y given b y Lemma 1.6; for simplicit y w e tak e κ t = 2 = { 0 , 1 } for eac h t ∈ X × 2 κ . Let M = { M ξ : ξ < 2 κ } = P ( X ) , and define K := f M as in Definition 2.9. W e will sho w that U := T KI D is as required. (a) The equalit y ∆( X, T KI D ) = ∆( X , T ) is giv en b y Corollar y 2.6, while S ( X , T KI D ) = S ( X, T ) is immediate from Lemma 2.7 (using t he regularit y of S ( X, T ) and the fact that κ t < ω < ω + ≤ S ( X, T ) for each t ∈ Z × 2 κ ). (b) According to Corolla r y 2.6(b), the disjoint sets D n ( n < ω ) a r e dense in ( X , T KI D ). (c) and (d) Supp ose there is a family E of pairwise disjoin t dense subsets of ( X , T KI D ) suc h that |E | = S ( X, T ). Note then that for some E ∈ E w e ha v e in t ( D n , T KI D ) ( D n ∩ E ) = ∅ for eac h n < ω . (*) (Indeed otherwise w e ma y argue as in [22](2.3), [7], [8](3.1(c)): c ho osing for eac h E ∈ E some n ( E ) < ω suc h that in t ( D n ( E ) , T KI D ) ( D n ( E ) ∩ E ) 6 = ∅ , w e ha ve f rom Lemma 2.7 and the regularity of S ( X , T ) = S ( X , T KI D ) that some ( fixed) n < ω satisfi es 10 in t ( D n , T KI D ) ( D n ∩ E ) 6 = ∅ for S ( X , T KI D )-man y E ∈ E ; that giv es S ( D n , T KI D ) > S ( X , T K I D ), whic h is impo ssible since D n is dense in ( X, T KI D ).) Then c ho o sing E ∈ E as in (*), w e hav e from Theorem 2.11(b) that E ∈ K , so E is closed and discrete in the cro wded space ( X, T K I D ) b y Lem ma 2.5((a) and (b)). This con tradicts the densit y of E in ( X , T KI D ).  Remark 3.2 The ch oice κ t < κ for all t ∈ X × 2 κ in (the pro of of ) Theorem 3.1 is essen tial. If κ t = κ is p ermitted for some t then the refinemen t U = T KI D satisfies conditions (b) and (c), but as noted in the first paragraph of the pro of of Lemm a 2.7 w e w ould no w hav e S ( X, T K I D ) = κ + > S ( X, T ). As is indicated in its pro of , Theorem 3.1 is of intere st o nly when the giv en space ( X , T ) is maximally resolv able. So view ed, the case κ ′ = S ( X , T ) of the follow ing result (cf. item ( i = 2) of our Abstract) strengthens a nd impro v es Theorem 3.1. Theorem 3.3 L et X = ( X, T ) b e a cr owde d, ma x imal ly r esolvable T ychonoff sp ac e and let κ ′ b e a r e gular c ar dinal such that S ( X, T ) ≤ κ ′ ≤ | X | = ∆( X , T ) = κ . Then ther e is a T ychonoff r efinement U of T such that (a) S ( X , U ) = κ ′ and ∆( X, U ) = ∆( X, T ) = κ ; (b) ( X, U ) is τ -r esolvable for e ach τ < κ ′ ; and (c) ( X, U ) is not κ ′ -r esolvab le. Pro of. [Being κ -resolv able, the space ( X, T ) is sure ly κ ′ -resolv able, so in this case the top o logy U will of ne cessit y b e a strict refinemen t of T .] Let Λ be the set of all cardinals τ suc h that 2 ≤ τ < κ ′ , and let { κ t : t ∈ T = X × 2 κ } list the elem ents of Λ with eac h τ ∈ Λ a pp earing 2 κ -man y times. F or τ ∈ Λ se t T ( τ ) := { t ∈ T : κ t = τ } . According to Lemma 1.6, there is a strong small-set-separating family κ -indep endent family I = {I t : t ∈ X × 2 κ } of partitio ns of κ whic h r esp ects the partition { T ( τ ) : τ ∈ Λ } of T . W e note that κ ′ = sup t ∈ T κ + t . Let D = { D n η : n < ω , η < κ } b e a dense partition of ( X , T ), and as usual set D n := S η<κ D n η . T ak e K as in Theorem 2.11 and set U := T KI D (with Z = X ). W e sho w that U is as required. (a) The equalities ∆( X , T KI D ) = ∆( X , T ) and S ( X , T KI D ) = κ ′ are given b y Corollary 2.6(c) and Lemma 2.7, resp ectiv ely . (c) The a rgumen t sho wing that the space ( X , T KI D ) of Theorem 3.1(c) is not S ( X , T K I D )-resolv able ( i.e., is not κ ′ -resolv able) a pplies here ve rb atim to pro ve (c). (b) Let A = { A n : n < ω } b e an arbitrary countable dens e partit io n o f the space ( X , T KI D ). Fix τ < κ ′ , let t ( n ) ( n < ω ) b e a f aithfully indexed sequence fro m X × 2 κ suc h that κ t ( n ) = τ for eac h n < ω , and for n < ω and α < τ set E α n := W α t ( n ) \ S k ps( X ) is maximally resolv able. That theorem w as strengthened in tw o w ays in [26]: No separation h yp o t hesis on X is required, and maximal res olv abilit y of X is es ta blished assuming only ∆( X ) ≥ ps( X ). Our pro of of Theorem 3.3 rests on the conv entions of Section 2, and uses crucially the (strong) h yp othesis that ( X , T ) is maximally resolv able. Th at h yp othesis can b e w eak ened to the ass umption tha t ( X, T ) is κ ′ -resolv able, with κ ′ regular and S ( X , T ) ≤ κ ′ ≤ | X | = ∆( X , T ) = κ , pro vided that the equalit y 2 κ ′ = 2 | X | is assumed. Indeed the argument giv en in the pro o f of Theorem 3.3 shows that U := T KI D has prop erties 12 (a), ( b) and (c), with D = { D n η : n < ω , η < κ ′ } a dense partition of ( X, T ), with I = {I t : t ∈ X × 2 κ ′ } a strong small-set-separating, κ ′ -indep enden t family of partitions of κ ′ , and with K = f M as in Definition 2.9 with Z = X . W e do not k now in ZFC whether the h yp o t hesis of Theorem 3.3 can b e w eak ened. Sp ecifically we ask: Question 3.5 Let X = ( X , T ) b e a crowded Ty chonoff space and let κ ′ b e a regular cardinal suc h that S ( X , T ) ≤ κ ′ < | X | = ∆( X , T ) = κ and ( X , T ) is τ -resolv able for eac h τ < κ ′ . Must there then exist, in ZF C, a T ych ono ff refinemen t U of T suc h that (a) S ( X, U ) ≤ κ ′ (p erhaps ev en: S ( X , U ) = S ( X, T )) and ∆( X, U ) = ∆( X , T ) = κ ); (b) ( X , U ) is τ -resolv able for eac h τ < κ ′ ; and (c) ( X , U ) is not κ ′ -resolv able? Of course, Question 3.5 is of inte rest only if ( X , T ) is itself κ ′ -resolv able, since otherwise U := T w ould b e as required. Next w e pro v e item ( i = 3) of the Abstract fo r the case | X | = ∆( X ). Theorem 3.6 L et X = ( X , T ) b e a cr owde d, maximal ly r esolvable T ychono ff sp ac e with S ( X , T ) ≤ | X | = ∆( X, T ) = κ . Then ther e is a T ycho noff r efinement U of T such that (a) S ( X , U ) = S ( X, T ) and ∆( X, U ) = ∆( X, T ) ; (b) ( X, U ) is maximal ly r esolvable; and (c) ( X, U ) is not extr ar eso lvable. Pro of. W e in vok e the conv en tions of 2.2 and 2.4, no w taking τ = κ . Let D = { D γ η : η < κ, γ < κ } be a faithfully indexe d dense pa r t ition of ( X , T ), and se t D γ := S η<κ D γ η for γ < κ . Let I = {I t : t ∈ X × 2 κ } b e a κ -indep endent family of partitions I t of X with the strong small-set-separating prop ert y; for simplic ity we tak e κ t = 2 = { 0 , 1 } for each t ∈ X × 2 κ . Let M = { M ξ : ξ < 2 κ } = P ( X ), and define K := f M as in D efinition 2.9 (taking Z = X ). W e w ill sho w that U := T KI D is as required. (a) The equalit y ∆( X, T KI D ) = ∆( X , T ) is giv en b y Corollar y 2.6, while S ( X , T KI D ) = S ( X, T ) is immediate from Lemma 2.7 (using t he regularit y of S ( X, T ) and the fact that κ t < ω < ω + ≤ S ( X, T ) for each t ∈ Z × 2 κ ). (b) According to Corollary 2 .6(b), the disjoint sets D γ ( γ < κ ) are dense in ( X , T KI D ). (c) Supp ose there is a family E of dense subsets o f ( X, T KI D ), with |E | = κ + , suc h that eve ry tw o elemen ts of E hav e in tersection whic h is no where dens e in ( X, T KI D ). W e claim that, m uch as in the proo f of Theorem 3.1(c), there is E ∈ E suc h that in t ( D γ , T KI D ) ( D γ ∩ E ) = ∅ for eac h γ < κ . (*) F or if (* ) fails then some (fixed) γ < κ satisfies in t ( D γ , T KI D ) ( D γ ∩ E ) 6 = ∅ for κ + -man y E ∈ E , and then s ince S ( D γ , T KI D ) = S ( X , T K I D ) = S ( X , T ) ≤ κ there are distinct E , E ′ ∈ E suc h that ∅ 6 = [in t ( D γ , T KI D ) ( D γ ∩ E )] ∩ [in t ( D γ , T KI D ) ( D γ ∩ E ′ )] = in t ( D γ , T KI D ) ( D γ ∩ E ∩ E ′ ). Then with T KI D -op en U ⊆ X c hosen so that D γ ∩ U = int ( D γ , T KI D ) ( D γ ∩ E ∩ E ′ ) w e ha ve 13 ∅ 6 = U ⊆ cl ( X, T KI D ) U = cl ( X, T KI D ) ( D γ ∩ U ) = cl ( X, T KI D ) in t ( X, T KI D ) ( D γ ∩ E ∩ E ′ ) ⊆ cl ( X, T KI D ) ( E ∩ E ′ ), con trary to the fact that E ∩ E ′ is now here dense in ( X , T KI D ). Thus (*) is established. Then, choosing E ∈ E as in (*), w e ha v e f rom Theorem 2 .11(b) (applied to the set M ξ = E ) t ha t E ∈ K = f M , so b y Lemma 2.5((a) a nd (b)) the set E is clos ed and discrete in the c rowded space ( X , T K I D ). This contradicts the density of E in ( X , T KI D ).  W e turn next to establishing items ( i = 4 ) and ( i = 5) o f the Abstract for the case | X | = ∆( X ). As e xp ected, refinemen ts of the form U = T KI D pla y a cen tr a l role; it is necessary only to tailor in eac h case the sp ecifics of the families K , I , and D to the task a t hand. But in Theorem 3.10 the pro ces s is iterated: a first expansion T ′ ⊇ T satisfies n wd( X , T ′ ) = κ , a second expansion T ′′ ⊇ T ′ is maximally resolv able but not extraresolv able, and a final expansion (of the fo rm T ′′I D , not T ′′ KI D ) has all required prop erties. F or the pro ofs of (the case | X | = ∆( X ) of ) items ( i = 4) and ( i = 5) of the Abstract, w e need tw o preliminary lemmas. A w eak v ersion of L emma 3.7 is prov ed in our w ork [7 ](3.9 ). A strictly com binatoria l proof exists, but it is length y; w e giv e inste ad an argumen t whic h uses the topolo g ical constructions already at our disp osal. Lemma 3.7 L et τ ≥ ω . Ther e exist families A = {A ξ : ξ < 2 τ } and S er ⊆ P ( τ ) such that (i) A is a τ -indep endent family of p artitions of τ with the str ong smal l-s e t-se p ar ating pr op erty, with e ach A ξ ∈ A of the form A ξ = { A 0 ξ , A 1 ξ } ; (ii) |S er | = 2 τ ; (iii) if n < ω and S, S 1 , S 2 , . . . S n ar e distinct elements of S er and A = T ξ ∈ F A f ( ξ ) ξ with F ∈ [2 τ ] <ω and f ∈ { 0 , 1 } F , then | A ∩ ( S \ ( S 1 ∪ S 2 ∪ . . . ∪ S n )) | = τ ; and (iv) if S, S ′ ∈ S er with S 6 = S ′ then (a) for e ach x ∈ τ \ ( S ∩ S ′ ) ther e ar e infinitely many ξ < 2 τ such that x ∈ A 1 ξ and S ∩ S ′ ⊆ A 0 ξ ; and (b) for e ach x ∈ S ∩ S ′ ther e ar e infinitely many ξ < 2 τ such that ( S ∩ S ′ ) ∩ A 1 ξ = { x } . Pro of. Let J ∪ L ∪ {D } b e a τ -indep enden t fa mily of part it io ns of τ , where J = {J ξ : ξ < 2 τ } is c hosen (as in Theorem 1.9) so that the space Y = ( Y , T ) := e J [ τ ] ⊆ K := { 0 , 1 } J = { 0 , 1 } 2 τ has prop erties (a), (b), (c) and (d) of Theorem 1.9. W e tak e |J | = |L| = 2 τ , sa y J = {J ξ : ξ < 2 τ } and L = {L ζ : ζ < 2 τ } , and w e tak e eac h J ξ ∈ J of the f o rm J ξ = { J 0 ξ , J 1 ξ } and each L ζ ∈ L of the form L ζ = { L 0 ζ , L 1 ζ } . W e write D = { D γ η : γ < τ , η < τ } . The families A and S er will b e defined with the help of a suitable expansion T K I D of T . The fa mily D has already b een defined, and for I w e ch o ose an arbitrary τ - indep enden t family I = {I t : t ∈ Y × 2 τ } of partitions of τ with the strong small- set-separating prop ert y , sa y with eac h I t of the fo r m I t = { I 0 t , I 1 t } . F or K , first let K ′ := { T ζ ∈ F L 0 ζ : | F | > 1 , F ∈ [2 τ ] <ω } and let K b e the set of sets of the form S i 1 suc h that K = S i ∆( X , U ) = ∆( X , T ′ )) witne ssing the strong e xtraresolv a bilit y of ( X , U ) w ould witness the strong extraresolv abilit y of ( X , T ′′ ), contrary to the fact that ( X , T ′′ ) is not (ev en) extraresolv a ble.  4 The Ge neral C ase The five principal res ults prov ed in Section 3 require, in additio n to the essen tial o ve rarch- ing h yp othesis S ( X , T ) ≤ ∆( X , T ), also the artificial condition | X | = ∆( X , T ). Since for eac h of those fiv e results it is ess entially the same argument whic h allow s us to pass fro m the sp ecial case ( | X | = ∆( X, T )) t o the unrestricted case ( | X | is arbitrary), we cor r al all fiv e of the general r esults in t o one extended statemen t. Theorem 4.2, t hen, duplicates the essen tials of our Abstract. Lemma 4.1 L et ( X, T ) b e a cr owde d T ycho noff sp ac e. F or ∅ 6 = U ∈ T ther e is V ∈ T such that K := cl ( X, T ) V satisfi e s V ⊆ K ⊆ U and ∆( U ) = ∆( K ) = | K | . Pro of. Cho o se W ∈ T such that W ⊆ U and | W | = ∆( U ), a nd choose V ∈ T so that V 6 = ∅ and V ⊆ K := cl ( X, T ) V ⊆ W .  Theorem 4.2 L et ( X , T ) b e a cr owde d, max i m al ly r eso lvable T ychonoff sp a c e such that S ( X , T ) ≤ ∆( X, T ) = κ . T hen ther e ar e T ychonoff e x p ansions U i ( 1 ≤ i ≤ 5 ) of T , with ∆( X, U i ) = ∆( X, T ) an d S ( X, U i ) ≤ ∆( X, U i ) , such t ha t ( X , U i ) is: ( i = 1 ) ω -r esolvable but no t maxima l ly r es o lvable; ( i = 2 ) [if κ ′ is r e gular, with S ( X , T ) ≤ κ ′ ≤ κ ] τ -r esolvable for al l τ < κ ′ , but not κ ′ -r esolvab le; ( i = 3 ) maximal ly r esolvable, but not extr ar esolv a ble; ( i = 4 ) extr ar esol v able, but not maxim a l ly r esolvable ; ( i = 5 ) maximal ly r esolvable and extr ar esolva ble, but not str ongly extr ar esolvable. Pro of. (Recall our frequen tly used conv en tion that when ( X , T ) is a space and Y ⊆ X , the sym b ol ( Y , T ) denotes the set Y with the top ology inherited from ( X, T ).) Using Lemma 4.1 (with U = X ), c ho ose a regular-closed set X ′ ⊆ X suc h that S ( X ′ , T ) ≤ S ( X, T ) ≤ ∆( X, T ) = ∆( X ′ , T ) = | X ′ | = κ . The definition of the top ologies U i for i = 1 , 2 , 3, and the v erification that they are as required, will b e straigh tforw ard. W e discuss these first, lea ving the cases ( i = 4 , 5) for treatmen t later in the pro of. The space ( X ′ , T ) satisfies the h yp ot heses of Theorems 3.1, 3.3, 3.6, so there are T yc honoff expansions U ′ i ( i = 1 , 2 , 3) of T on X ′ satisfying their resp ectiv e conclusions. Let U i ( i = 1 , 2 , 3) be the t o p ology on X for whic h ( X ′ , U ′ i ) and ( X \ X ′ , T ) are op en-a nd- closed subspaces o f ( X, U i ). It is easily seen t ha t ( X , U i ) is a T ych ono ff space. F urther w e 20 ha v e T ⊆ U i , since if U ∈ T then U ∩ X ′ is op en in ( X ′ , T ), hence in ( X ′ , U ′ i ), hence in ( X ′ , U i ), and U ∩ ( X \ X ′ ) is open in ( X \ X ′ , T ) = ( X \ X ′ , U i ). F or i = 1 , 2 , 3 w e hav e, using ∆( X ′ , U ′ i ) = ∆( X ′ , T ) ≤ ∆( X \ X ′ , T ), tha t ∆( X, U i ) = min { ∆( X ′ , U i ) , ∆( X \ X ′ , U i ) } = ∆( X ′ , T ) = ∆( X, T ) = κ . F urther f o r i = 1 , 3 w e ha v e, using S ( X ′ , U ′ i ) = S ( X ′ , T ), that S ( X , U i ) = S ( X ′ , U i ) + S ( X \ X ′ U i ) = S ( X ′ , T ) + S ( X \ X ′ , T ) = S ( X , T ), while for i = 2 we ha v e S ( X , U 2 ) = S ( X ′ , U 2 ) + S ( X \ X ′ , U 2 ) = κ ′ + S ( X \ X ′ , U 2 ) = κ ′ + S ( X \ X ′ , T ) = κ ′ . W e v erify the required (non- ) resolv ability pr o p erties of the spaces ( X , U i ) for i = 1 , 2 , 3. In eac h case, ( X, U i ) is the union of t w o disjoin t op en-and- closed subspaces, namely ( X ′ , U i ) and ( X \ X ′ , U i ) = ( X \ X ′ , T ). Whe n i = 1, these are b oth ω -resolv able; when i = 2, b oth are τ -resolv able for eac h τ < κ ′ ; when i = 3, b oth are κ -resolv able. Th us ( X , U 1 ) is ω -resolv able; ( X, U 2 ) is τ -resolv able for all τ < κ ′ ; and ( X , U 3 ) is κ -resolv able (i.e., is maximally resolv able). Since ( X ′ , U ′ 1 ) = ( X ′ , U 1 ) is op en in ( X, U 1 ) and is not ∆( X, T )-resolv a ble, surely ( X , U 1 ) is not ∆( X, T )-resolv able, i.e., is not ∆( X ′ , U 1 )-resolv able. The space ( X, U 2 ) is not κ ′ -resolv able, since its o p en subspace ( X ′ , U ′ 2 ) = ( X ′ , U 2 ) is not κ ′ -resolv able. The space ( X, U 3 ) is no t extraresolv able, since its op en subs pace ( X ′ , U ′ 3 ) = ( X ′ , U 3 ) is not ex tra r esolv able ( and satisfies ∆( X ′ , U ′ 3 ) = ∆( X, U 3 )). W e turn to the cases ( i = 4 , 5). Let V ⊆ T b e c hosen maximal with resp ect to the prop erties { cl ( X, T ) V : V ∈ V } is pairwise disjoint, and | V | = | cl ( X, T ) V | = ∆( V ) for eac h V ∈ V . W e write V = { V β : β < α } and X ′ β := cl ( X, T ) V β , the indexing ch osen with V 0 and X ′ 0 = X ′ as in the first part of this pro of: | X ′ 0 | = ∆( X ′ 0 , T ) = ∆( X, T ). The space ( X ′ 0 , T ) satisfies S ( X ′ 0 , T ) ≤ S ( X, T ) ≤ ∆( X, T ) = ∆( X ′ 0 , T ) = | X 0 | = κ , so by Theorems 3.9 and 3.10 there are Tyc honoff refinemen ts U ′ 0 , 4 and U ′ 0 , 5 of ( X ′ 0 , T ), with S ( X ′ 0 , U ′ 0 , 4 ) = S ( X ′ 0 , U ′ 0 , 5 ) = S ( X ′ 0 , T ) and ∆( X ′ 0 , U ′ 0 , 4 ) = ∆( X ′ 0 , U ′ 0 , 5 ) = ∆( X ′ 0 , T ) = κ , suc h that ( X ′ 0 , U ′ 0 , 4 ) is extraresolv able, but not maxim ally resolv able; and ( X ′ 0 , U ′ 0 , 5 ) is maximally resolv able and extraresolv able, but not strongly extrare- solv able. F or 0 < β < α the spaces ( X ′ β , T ) satisfy S ( X ′ β , T ) ≤ S ( X, T ) ≤ κ = ∆( X , T ) ≤ ∆( X ′ β , T ) = | X ′ β | . By Theorem 3.8, taking τ = κ β := | X ′ β | there, there are for 0 < β < α T yc honoff expansions U ′ β of ( X ′ β , T ) suc h that S ( X ′ β , U ′ β ) = S ( X ′ β , T ) and ∆( X ′ β , U ′ β ) = ∆( X ′ β , T ), and ( X ′ β , U ′ β ) is κ β -resolv able and 2 κ β -extraresolv able. Then since κ ≤ κ β , the space ( X ′ β , U ′ β ) is κ -resolv able and 2 κ -extraresolv able. No w fo r ( i = 4 , 5) w e define U i to b e the smallest top olog y on X suc h that 21 (1) T ⊆ U i , (2) ( X ′ 0 , U ′ 0 ,i ) is op en-a nd-closed in ( X , U i ), and (3) eac h space ( X ′ β , U ′ β ) (with 0 < β < α ) is open- a nd-closed in ( X , U i ). T o see that ( X, U i ) is a T yc honoff space, it is enough to note that if x ∈ S β <α X ′ β , sa y x ∈ X ′ β , then X ′ β is an op en T ychonoff neigh b orhoo d of x in ( X , U i ); while if x / ∈ S β <α X ′ β , then the T -op en neigh b orho o ds of x remain bas ic a t x in ( X, U i ) (so if x ∈ U ∈ U i then t here is a U i -con tinuous (ev en, T - con tin uous) real-v alued f unction f on X suc h that f ( x ) = 0 and f = 1 on X \ U ). F or β < α w e hav e ∆( X ′ β , U i ) = ∆( X ′ β , U ′ β ) = ∆( X ′ β , T ) ≥ ∆( X ′ 0 , T ), so ∆( X , U i ) = min β <α ∆( X ′ β , U i ) = ∆( X ′ 0 , T ) = ∆( X , T ) = κ . W e v erify for ( i = 4 , 5 ) that S ( X, U i ) ≤ ∆( X , U i ). F or a cellular family W ⊆ U i and β < α let W ( β ) := { W ∩ X ′ β : W ∈ W , W ∩ X ′ β 6 = ∅} . Then W ( β ) is a cellular family b y Lemma 4.1. The set S β <α X ′ β is dense in ( X , T ), so W = S β <α W ( β ), so |W | ≤ Σ β <α |W ( β ) | with eac h |W ( β ) | < S ( X ′ β , U ′ β ) = S ( X ′ β , T ) ≤ S ( X, T ). Since α < S ( X , T ) a nd S ( X, T ) is r egula r , w e ha v e |W | < S ( X, T ). It follo ws that S ( X , U i ) ≤ S ( X, T ) ≤ ∆( X , T ) = ∆( X , U i ). It remains to v erify that the spaces ( X , U 4 ) and ( X , U 5 ) ha v e t he required (non-) resolv abilit y prop erties. Eac h space ( X ′ β , U 4 ) is op en in ( X, U 4 ), with S β <α X ′ β dense in ( X , U 4 ). Each space ( X ′ β , U 4 ) is extraresolv able (b y Theorem 3.9(b) for β = 0, by Theorem 3.8 for 0 < β < α ), so for e ach β < α the re is a family E β = { E β ( η ) : η < κ + } of den se subse ts of ( X ′ β , U 4 ) suc h that E β ( η ) ∩ E β ( η ′ ) is now here dense in ( X ′ β , U 4 ) whenev er η < η ′ < κ + . Then with E ( η ) := S β <α E β ( η ), t he family { E ( η ) : η < κ + } witnesses the extraresolv abilit y of ( X , U 4 ). The s pace ( X , U 4 ) is not maximally resolv able (i.e., is not κ -resolv able), how eve r, since its o p en subspace ( X ′ 0 , U 4 ) = ( X ′ 0 , U 4 , 0 ) is not κ -resolv a ble. The space S β <α X ′ β is o p en and dense in ( X , U 5 ), with eac h ( X ′ β , U 5 ) op en and κ -resolv able and 2 κ -extraresolv able, so ( X , U 5 ) is κ -resolv able (i.e., maximally resolv able) and ex tra r esolv able. Eac h set K ∈ [ X ′ 0 ] <κ is closed and discrete in ( X ′ 0 , U 0 , 5 ) = ( X ′ 0 , U 5 ), so nw d ( X ′ 0 , U 5 ) = κ . Thus an y family of sets dense in ( X , U 5 ) witnessing the strong extraresolv abilit y of ( X , U 5 ) w ould trace on ( X ′ 0 , U 5 ) to a family witnessing strong ex- traresolv ability there.  5 Some Qu e stions Both our result cited fro m [8] in Remark 3.4(b) (where S ( X ) > | X | ) and its sequel in Theorem 3.3(b) (where S ( X ) ≤ | X | ) s how that in some case s ω -resolv a bilit y suffic es to guaran tee τ -resolv ability for many larger τ . Our metho ds app ear insufficien tly delicate, ho w ev er, to resp ond to the following question. Question 5.1 Let ( X, T ) b e a n ω -resolv able Ty chonoff space suc h that S ( X , T ) ≤ ∆( X, T ). Must ( X , T ) b e τ -resolv able for ev ery τ < S ( X , T )? 22 Question 5.2 Let X = ( X , T ) b e a dense, ω -resolv able subspace of the space ( D ( κ )) 2 κ suc h that | X | = ∆( X ) = κ . [Then S ( X ) = κ + , and X is κ -resolv able, i.e., maximally resolv able, according to o ur result [8](4 .2 ).] Do es X a dmit a T yc honoff refinemen t U (necessarily with S ( X, U ) = κ + ) suc h that ∆( X , U ) = ∆( X, T ), and ( X , U ) is ω -r esolv able but not maximally resolv able? Alw a ys? Sometimes? Nev er? Remarks 5.3 (a) Theorem 3.1 sheds no ligh t on Question 5.2 , since the h yp othesis S ( X , T ) ≤ ∆( X, T ) is lacking. (b) The expans ion U of T requested in Question 5.2, if it exists , cannot be of the kind constructed in this pap er. More sp ecifically: There can b e no family W ⊆ P ( X ) suc h that (i) | U ∩ W | = κ for eac h W ∈ W and ∅ 6 = U ∈ T , (ii) U is the smallest top ology on X con taining T and W , and (c) each W ∈ W is U -clop en. F or according to the argumen t outlined in Discussion 1.5, a space ( X, U ) arising in that wa y will em b ed as a dense subspace of ( D ( κ )) I (with | I | = w ( X , U )), hence if ω -resolv able is necessarily κ -resolv able. (c) Man y additional questions relating to (ir)resolv a bility , together with extens ive bibliographic citations, are recorded in the “Problems” article o f P av lov [30]. Remark 5.4 The reader will ha ve no difficult y using the methods of this pap er to estab- lish the fo llowing result: (*) L et ( X , T ) b e a cr owde d, maxim a l ly r e s olvable T ycho n off sp ac e with S ( X , T ) ≤ ∆( X , T ) = κ . The n for (fixe d) n < ω ther e is a T ychonoff exp ansion U of T s uch t ha t ( X , U ) is n -r esolvable but not ( n + 1) -r esolvab le. (Indeed, reducing as in Theorem 4 .2 to the cas e | X | = ∆( X, T ), it is en oug h to b egin with a dense pa r t it ion { D k η : k < n, η < κ } of ( X , T ), a strong small-set-separating κ -indep enden t family I = {I t : t ∈ X × 2 κ } of κ with eac h I t = { I 0 t , I 1 t } , and with the family K = { K ξ : ξ < 2 κ } define d as in Theorem 2.11. Then the relation X = S k

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