Noetherian types of homogeneous compacta and dyadic compacta

NOETHERIAN TYPES OF HOMOGENE O US COMP A CT A AND D Y ADIC COMP A CT A DA VID MILOVICH Abstract. The Noetherian t yp e of a space is the least κ such that it has a base that is κ -like with respect to cont ainment. Just as all known homogeneo us compacta hav e cellular i t y at most c , they satisfy si milar upper b ounds in terms of No etherian t yp e and related cardinal f unctions. W e prov e these and many other r esults ab out the se cardinal functions. F or example, ev ery homoge neous dy adic compactum has No etherian t yp e ω . Assuming GCH, every p oin t in a homogene ous compactum X has a local base that is c ( X )-like with resp ect to con tainmen t. I f eve ry p oint in a compactum has a well-quasiordered lo cal base, then some p oint ha s a countab le local π - base. 1. Introduction V a n Douw en’s P roblem (see Kunen [1 2]) asks whether there is a ho mogeneous compactum of c e llularity exceeding c . (See Enge lk ing [6], Juh´ asz [10], a nd Kunen [13] for all undeﬁned terms. In pa rticular, reca ll that w ( · ), π ( · ), χ ( · ), π χ ( · ), d ( · ), c ( · ), and t ( · ) resp ectively denote w eight, π -weight , character, π -character, densit y , cellu- larity , a nd tightness of to p olo gical spaces.) A homog eneous compactum of cellular - it y c ex ists by Maur ice [15], but v an Douw en’s Problem re ma ins o p en in all models of ZFC. Deﬁnition 1. 1 . W e say that a homogeneous compactum is exc eptional if it is no t homeomorphic to a pro duct of dyadic compacta and ﬁrst co un table co mpacta. By Arhange l ′ ski ˘ ı’s Theorem, ﬁrst coun table spaces have size at mos t c ; dyadic compacta are ccc. Since the ce llula rity of a pro duct space eq uals the supremum of the cellular ities of its ﬁnite subpro ducts (see p. 107 o f [10]), all no ne x ceptional homogeneous compacta hav e cellularity a t most c . T o the b est of the author ’s knowl- edge, there are only t wo class es o f examples of ex ceptional homogeneo us compacta (see [16]); these tw o kinds of spac e s hav e cellularities ω and c . W e inv estiga te several ca rdinal functions deﬁned in terms or der-theore tic bas e prop erties. J ust like cellular it y , these functions hav e upp er b ounds when restricted to the class of known homogeneo us compacta. Moreov er, GCH implies that one of these functions is a low er b ound on cellular ity when restric ted to homogeneo us compacta. Deﬁnition 1 .2. Giv en a cardina l κ , deﬁne a p os et to b e κ - like ( κ op - like ) if no element is ab ov e (b elow) κ -many elemen ts. Deﬁne a p os et to b e almost κ op - like if it has a κ op -like dense subset. Date : May 15, 20 07. Support pro vided by an NSF graduate fello wship. 1 2 DA VID MILO VICH In the context of families o f subsets of a top o lo gical space, we will always im- plicitly o r der by inclus io n. Deﬁnition 1.3. Given a space X . let the No etherian typ e of X , or N t ( X ), b e the least κ ≥ ω s uch that X has a ba s e that is κ op -like. Analogously deﬁne No etherian π -typ e in terms of π -bases and deno te it by π N t . Given a subset E of X , let the lo c al No et herian typ e of E in X , o r χN t ( E , X ), b e the lea st κ ≥ ω such that there is a κ op -like neighborho o d base of E . Given p ∈ X , let the lo ca l No etherian t ype of p , or χN t ( p, X ), be χN t ( { p } , X ). Let the lo cal N o etheria n t yp e o f X , or χN t ( X ), be the supremum o f the lo cal No etherian t yp es of its p oints. Let the c omp act No etherian typ e of X , or χ K N t ( X ), b e the supremum of the lo cal No e therian types of its compact subsets. W e ca ll N t , π N t , χN t , and χ K N t No etherian c ar dinal functions . No etherian t yp e and No etheria n π - t yp e were introduced b y P eregudov [19]. Pr e- ceding this introduction a re several pap ers by Peregudov, Shapirovski ˘ ı and Ma- lykhin [14, 17, 18, 20] ab out min { N t ( · ) , ω 2 } and min { π N t ( · ) , ω 2 } (using diﬀerent terminologies ). Also, Dow and Zhou [4] show ed that β ω \ ω ha s a p oint with lo cal No etherian type ω . Observ ation 1.4. Every known homo gene ous c omp actum X satisﬁes the fol lowing. (1) N t ( X ) ≤ c + . (2) π N t ( X ) ≤ ω 1 . (3) χN t ( X ) = ω . (4) χ K N t ( X ) ≤ c . W e justify this obser v a tio n in Section 2, exce pt that we p ostp one the case of ho- mogeneous dy adic compacta to Sec tion 3, where w e in vestigate No etheria n cardinal functions o n dyadic co mpacta in genera l. The results relev ant to Obse r v a tio n 1.4 are summarized b y the following theorem. Theorem 1.5. Supp ose X is a dyadic c omp actum. Then π N t ( X ) = χ K N t ( X ) = ω . Mor e over, if X is homo gene ous, then N t ( X ) = ω . Also in Section 3, we generalize the ab ov e theorem to contin uous imag e s of pro ducts of co mpacta with bo unded weigh t; we also pr ov e the following. Theorem 1.6 . The class of No etherian typ es of dyadic c omp acta includes ω , ex- cludes ω 1 , includes al l singular c ar dinals, and includes κ + for al l c ar dinals κ with unc ountable c oﬁnality. Section 4 inv estigates to what extent a technical prop erty of free b o olean alg ebras that is cruc ia l to Section 3 holds in o ther bo o lean alg ebras. In Section 5, we prov e several r e sults ab out the lo ca l No etheria n types of a ll homo geneous co mpacta, known and unknown, including the following theor em. Theorem 1.7 (GCH) . If X is a homo gene ous c omp actu m, then χN t ( X ) ≤ c ( X ) . 2. Obser ved upper bound s on No etherian cardinal functions First, we note so me very basic facts ab o ut No etheria n cardinal functions. Deﬁnition 2.1. Given a subset E of a pro duct Q i ∈ I X i and σ ∈ [ I ] <ω , we say that E has supp ort σ , or s upp( E ) = σ , if E = π − 1 σ π σ [ E ] and E 6 = π − 1 τ π τ [ E ] for all τ ( σ . NOETHERIAN TYPES OF HOMOGENEOUS COMP A CT A AND DY ADIC COMP ACT A 3 Theorem 2.2 . Given a p oint p and a c omp act su bset K of a pr o duct sp ac e X = Q i ∈ I X i , we have t he fol lowing r elations. N t ( X ) ≤ sup i ∈ I N t ( X i ) π N t ( X ) ≤ sup i ∈ I π N t ( X i ) χN t ( p, X ) ≤ sup i ∈ I χN t ( p ( i ) , X i ) χN t ( K, X ) ≤ sup σ ∈ [ I ] <ω χ K N t ( π σ [ K ] , π σ [ X ]) Pr o of. See Peregudo v [19] for a pro of of the ﬁrst relation. That pro of can b e easily mo diﬁed to demonstra te the next tw o rela tions. Let us prov e the last relatio n. F or each σ ∈ [ I ] <ω , set κ σ = χN t ( π σ [ K ] , π σ [ X ]) and let A σ be a κ op σ -like neigh b orho o d base of π σ [ K ]. F or ea ch σ ∈ [ I ] <ω , let B σ denote the set of sets of the form π − 1 σ U where U ∈ A σ and supp( U ) = σ . Note that if U ∈ A σ and supp( U ) ( σ , then there exists τ ( σ and V ∈ A τ such that π − 1 τ V ⊆ π − 1 σ U . Mo reov er, for an y minimal suc h τ , we ha ve π − 1 τ V ∈ B τ . Set B = S σ ∈ [ I ] <ω B σ . By co mpactness, B is a neighborho o d ba se of K . Moreov er, if σ, τ ∈ [ I ] <ω and B σ ∋ U ⊆ V ∈ B τ , then σ = s upp( U ) ⊇ supp ( V ) = τ ; hence, given U , there are at most (sup τ ⊆ σ κ τ )-many p ossibilities for V . Thus, B is (sup σ ∈ [ I ] <ω κ σ ) op -like as desired.  Question 2.3 . Do there exist spaces X and Y such that χ K N t ( X × Y ) exceeds χ K N t ( X ) χ K N t ( Y )? Lemma 2.4. Every p oset P is almost | P | op -like. Pr o of. Let κ = | P | and let h p α i α<κ enum era te P . Deﬁne a partial map f : κ → P as follows. Supp ose α < κ and w e hav e a partial map f α : α → P . If ra n f α is dense in P , then set f α +1 = f α . Otherwise, set β = min { δ < κ : p δ 6≥ q for a ll q ∈ ran f α } and let f α +1 be the sma llest map extending f α such that f α +1 ( α ) = p β . F or limit ordinals γ ≤ κ , set f γ = S α<γ f α . Then f κ is nonincreasing ; hence, r an f κ is κ op -like. Mor eov er, ran f κ is dense in P .  Theorem 2.5. F or any sp ac e X with p oint p , we have χN t ( p, X ) ≤ χ ( p, X ) and π N t ( X ) ≤ π ( X ) and N t ( X ) ≤ w ( X ) + and χ K N t ( X ) ≤ w ( X ) . Pr o of. The ﬁr s t tw o relations immediately follow from Lemma 2.4; the third r elation is trivial. F or the last relatio n, note that if K is a compact subset of X , then it has neighborho o d base of size at most w ( X ); apply Lemma 2.4.  Given Theorem 2 .2, justifying Observ ation 1.4 for N t ( · ), π N t ( · ), and χN t ( · ) amounts to justifying it for ﬁrst countable ho mogeneous compacta, dyadic homoge- neous compacta, and the tw o known kinds o f exceptional homogeneous co mpa cta. The ﬁrst countable case is the ea siest. By Arha ngel ′ ski ˘ ı’s Theo rem, ﬁrst countable compacta hav e w eight at most c , and therefore hav e Noether ian type at mo st c + . Moreov er, every p oint in a ﬁrs t count able space clearly has an ω op -like lo cal base. The only no ntrivial b ound is the o ne on Noetheria n π -type. F or that, the following theorem suﬃces. Deﬁnition 2.6. Give a space X , let π sw ( X ) denote the leas t κ such that X has a π -bas e A s uch that T B = ∅ for all B ∈ [ A ] κ + . 4 DA VID MILO VICH Theorem 2.7. If X is a c omp actum, then π N t ( X ) ≤ π sw ( X ) + ≤ t ( X ) + ≤ χ ( X ) + . Pr o of. Only the second r elation is nontrivial; it is a theorem o f Shapirovski ˘ ı [21].  F or dyadic homog e neous compac ta, Theorem 1.5 obviously implies Obser v a- tion 1 .4; we will prove this theorem in Section 3. Now c o nsider the tw o known classes ex ceptional ho mogeneous co mpacta. They are constructed by tw o tech- niques, reso lutions and a malgams. Fir st we consider the exceptional r esolution. Deﬁnition 2.8. Supp ose X is a space, h Y p i p ∈ X is a se q uence o f nonempty spaces , and h f p i p ∈ X ∈ Q p ∈ X C ( X \ { p } , Y p ). Then the r esolution Z of X at each p oint p into Y p by f p is deﬁned b y setting Z = S p ∈ X { p } × Y p and declaring Z to hav e weak est top ology s uc h that, for e very p ∈ X , op en neighborho o d U of p in X , and ope n V ⊆ Y p , the set U ⊗ V is op en in Z where U ⊗ V = ( { p } × V ) ∪ S q ∈ U ∩ f − 1 p V { q } × Y q . The resolutio n of c o ncern to us in constr uc ted by v an Mill [23]. It is a compactum with weigh t c , π -weigh t ω , and character ω 1 . Moreover, assuming MA + ¬ CH (or just p > ω 1 ), this space is homogeneous. (It is no t homogeneous if 2 ω < 2 ω 1 .) F or a pro of that this space is exceptio nal (as suming MA + ¬ CH), see [16]. Cle a rly , this space has suﬃciently small Noether ia n t yp e and π -type. W e just need to show that it has lo cal No etherian type ω . V an Mill’s space is a resolution of 2 ω at each p oint int o T ω 1 where T is the circle g roup R / Z . Notice tha t T is metriza ble. The following lemma proves that e very metr ic compactum has Noether ian type ω , a long with some results that will b e useful in Section 3. Lemma 2 . 9. L et X b e a metric c omp actum with b ase A . Then ther e exists B ⊆ A satisfying the fol lowing. (1) B is a b ase of X . (2) B is ω op -like. (3) If U, V ∈ B and U ( V , then U ⊆ V . (4) F or al l Γ ∈ [ B ] <ω , ther e ar e only ﬁnitely many U ∈ B such that Γ c ont ains { V ∈ B : U ( V } . Pr o of. Construct a sequenc e hB n i n<ω of ﬁnite subsets of A as follows. F or each n < ω , let E n be the union of the set of all sing letons in S m n , then diam V < diam U , in contradiction w ith U ( V . Hence, m < n ; hence, U ⊆ V . F or (1), let p ∈ X and n < ω , and let V b e the op en ball with r adius 2 − n and center p . Then we just nee d to show that there exists U ∈ B such that p ∈ U ⊆ V . Hence, we may ass ume { p } 6∈ B . Hence, p 6∈ E n +1 ; hence, there exists U ∈ B n +1 such that p ∈ U . Since diam U ≤ 2 − n − 1 , we hav e U ⊆ V . NOETHERIAN TYPES OF HOMOGENEOUS COMP A CT A AND DY ADIC COMP ACT A 5 F or (2), let n < ω and U ∈ B n . If U is a singleto n, then every superset o f U in B is in S m ≤ n B m . If U is no t a singleton, then U has diamater at least 2 − m for some m < ω ; whence, every sup ers e t of U in B is in S l ≤ m B l . F or (4), supp ose Γ ∈ [ B ] <ω and ther e exis t inﬁnitely many U ∈ B s uch that { V ∈ B : U ( V } ⊆ Γ. W e may assume Γ contains no singletons. Cho ose a n increasing sequence h k n i n<ω in ω such that, for a ll n < ω , there ex ists U n ∈ B k n such that { V ∈ B : U n ( V } ⊆ Γ. F or each n < ω , cho o se p n ∈ U n . Sin ce { U n : n < ω } is inﬁnite, we may choose h p n i n<ω such that { p n : n < ω } is inﬁnite. Let p be an accumulation po in t of { p n : n < ω } . Choo se m < ω s uc h that 2 − m < dia m V for all V ∈ Γ. Since p is not an isolated p oint, there exists W ∈ B m such that p ∈ W . Then W 6∈ Γ; hence, W do es not stric tly contain U n for any n < ω . Cho ose q ∈ W \ { p } s uch that W contains { x : d ( p, x ) ≤ d ( p, q ) } ; s e t r = d ( p, q ). Let B b e the o p en ball of r adius r/ 2 cen tered ab out p . Then there exists n < ω such that 2 − k n < r / 2 a nd p n ∈ B . Hence, diam U n < r / 2 a nd U n ∩ B 6 = ∅ ; hence, U n ⊆ W and q 6∈ U n ; hence, U n ( W , whic h is absurd. Ther efore, for each Γ ∈ [ B ] <ω , there are only ﬁnitely many U ∈ B such that { V ∈ B : U ( V } ⊆ Γ.  W e hav e N t (2 ω ) = N t ( T ω 1 ) = ω by Lemma 2.9 and Theo r em 2.2. Therefor e, the following theorem implies that v an Mill’s space ha s lo cal No ether ian type ω . Lemma 2.10 ([23]) . Supp ose X , h Y p i p ∈ X , h f p i p ∈ X , and Z ar e as in Deﬁn ition 2.8. Supp ose U is a lo c al b ase at a p oint p in X and V is a lo c al b ase at a p oint y in Y p . Then { U ⊗ V : h U, V i ∈ U × ( V ∪ { Y p } ) } is a lo c al b ase at h p, y i in Z . Theorem 2.11. S upp ose X , h Y p i p ∈ X , h f p i p ∈ X , and Z ar e as in Deﬁnition 2.8. Then χN t ( h p, y i , Z ) ≤ N t ( X ) χN t ( y , Y p ) for al l h p, y i ∈ Z . Pr o of. Set κ = N t ( X ) χN t ( y , Y p ). Let A be a κ op -like bas e of X and let B b e a κ op -like lo cal base at y in Y p ; we may assume Y p ∈ B . Set C = { U ∈ A : p ∈ U } . Set D = { U ⊗ V : h U, V i ∈ C × B } , whic h is a lo ca l base at h p, y i in Z by Lemma 2.10. If there exists U ⊗ V ∈ D such that U ∩ f − 1 p V = ∅ , then U ⊗ V is homeomor phic to V ; whence, χN t ( h p, y i , Z ) = χN t ( y , Y p ) ≤ κ . Hence, we may assume U ∩ f − 1 p V 6 = ∅ for all U ⊗ V ∈ D . It suﬃces to show that D is κ op -like. Suppose U i ⊗ V i ∈ D for a ll i < 2 and U 0 ⊗ V 0 ⊆ U 1 ⊗ V 1 . Then V 0 ⊆ V 1 and ∅ 6 = U 0 ∩ f − 1 p V 0 ⊆ U 1 ∩ f − 1 p V 1 . Since B is κ op -like, there ar e fewer than κ - many p ossibilities for V 1 given V 0 . Since A is a κ op -like base, there ar e fewer than κ - many p os s ibilities for U 1 given U 0 and V 0 . Hence, there are fewer than κ -ma n y p oss ibilities for U 1 ⊗ V 1 given U 0 ⊗ V 0 .  Deﬁnition 2.12. Let p denote the least κ for which some A ∈ [[ ω ] ω ] κ has the strong ﬁnite intersection prop erty but do es not hav e a nont rivia l pseudointersection. By a theo rem o f Bell[3], p is also the least κ for which there exist a σ -centered p oset P and a family D of κ - ma n y dense subse ts of P such that P do es not have a D -g e ne r ic ﬁlter. Deﬁnition 2.13 . Given a space X , let Aut( X ) denote the set of its autohomeo- morphisms. V a n Mill’s construction has b een generaliz ed by Har t and Ridderb os[8]. They show that o ne can pro duce an exce ptio nal homogeneous compactum with weight c and π -weight ω b y car efully resolving ea ch po int of 2 ω int o a ﬁxed space Y satisfying the following conditions. 6 DA VID MILO VICH (1) Y is a homog e ne o us compactum. (2) ω 1 ≤ χ ( Y ) ≤ w ( Y ) < p . (3) ∃ d ∈ Y ∃ η ∈ Aut( Y ) { η n ( d ) : n < ω } = Y . (4) If γ ω is a compac tiﬁca tion of ω and γ ω \ ω ∼ = Y , then Y is a retr act of γ ω . By Theorem 2.11, to s how that suc h res o lutions have lo ca l No etherian type ω , it suﬃces to show that every such Y has lo cal No etherian type ω . Theorem 2.16 will accomplish this. Theorem 2. 14. Supp ose X is a c omp actum and π χ ( p, X ) = χ ( q , X ) for al l p, q ∈ X . Then χN t ( p, X ) = ω for some p ∈ X . In p articular, if X is a homo gene ous c omp actum and π χ ( X ) = χ ( X ) , then χN t ( X ) = ω . The pro of of Theor em 2.14 will b e delayed until Section 5 . The following lemma is essentially a gener alization of a similar r e s ult of Juh´ asz[11]. Lemma 2.15 . S u pp ose X is a c omp actu m and ω = d ( X ) ≤ w ( X ) < p . Then ther e exists p ∈ X such that χ ( p, X ) ≤ π ( X ) . Pr o of. Let A b e a bas e of X of size at mo st w ( X ). Let B b e a π - base of X of size a t most π ( X ). F or each h U, V i ∈ B 2 satisfying U ⊆ V , cho ose a closed G δ -set Φ( U, V ) such that U ⊆ Φ( U, V ) ⊆ V . Then ran Φ, or dered by ⊆ , is σ -centered bec ause d ( X ) = ω . Since |A| < p , there is a ﬁlter G of ran Φ such that for all disjoint U , V ∈ A some K ∈ G s a tisﬁes U ∩ K = ∅ or V ∩ K = ∅ . Hence, ther e exists a unique p ∈ T G . Hence, p ha s pse udo character , and therefore character, a t most |G | , whic h is at most π ( X ).  Theorem 2.1 6 . If X is a homo gene ous c omp actum and ω = d ( X ) ≤ w ( X ) < p , then χN t ( X ) = ω . Pr o of. By Lemma 2.15, χ ( X ) ≤ π ( X ) = π χ ( X ) d ( X ) = π χ ( X ). Hence, by T he o - rem 2.14, χ N t ( X ) = ω .  Amalgams a r e deﬁned in [16] a s follows. Deﬁnition 2.17. Supp ose X is a T 0 space, S is a subbase of X such that ∅ 6∈ S , and h Y S i S ∈ S is a sequence of nonempty spaces. The amalgam Y of h Y S : S ∈ S i is deﬁned by setting Y = S p ∈ X Q p ∈ S ∈ S Y S and declaring Y to have the weakest top ology s uc h that, for each S ∈ S and op en U ⊆ Y S , the set π − 1 S U is op en in Y where π − 1 S U = { p ∈ Y : S ∈ dom p and p ( S ) ∈ U } . Deﬁne π : Y → X by { π ( p ) } = T dom p for all p ∈ Y . It is easily veriﬁed that π is contin uous. Theorem 2.18 . S upp ose X , S , h Y S i S ∈ S , and Y b e as in Deﬁnition 2.17 . Then we have the fol lowing r elations for al l p ∈ Y . N t ( Y ) ≤ N t ( X ) sup S ∈ S N t ( Y S ) π N t ( Y ) ≤ π N t ( X ) sup S ∈ S π N t ( Y S ) χN t ( p, Y ) ≤ χN t ( π ( p ) , X ) sup S ∈ dom p χN t ( p ( S ) , Y S ) Pr o of. W e will only prov e the ﬁrst rela tion; the pro ofs of the other s are almost ident ical. Set κ = N t ( X ) sup S ∈ S N t ( Y S ). Let A b e a κ op -like base of X . F or each NOETHERIAN TYPES OF HOMOGENEOUS COMP A CT A AND DY ADIC COMP ACT A 7 S ∈ S , let B S be a κ op -like base of Y S . Set C =  π − 1 U ∩ \ S ∈ dom τ π − 1 S τ ( S ) : τ ∈ [ F ∈ [ S ] <ω Y S ∈F B S \{ Y S } and A ∋ U ⊆ \ dom τ  . Then C is c learly a base o f Y . Let us show that C is κ op -like. Supp ose π − 1 U i ∩ T S ∈ dom τ i π − 1 S τ i ( S ) ∈ C for all i < 2 and π − 1 U 0 ∩ \ S ∈ dom τ 0 π − 1 S τ 0 ( S ) ⊆ π − 1 U 1 ∩ \ S ∈ dom τ 1 π − 1 S τ 1 ( S ) . Then U 0 ⊆ U 1 and do m τ 0 ⊇ do m τ 1 and τ 0 ( S ) ⊆ τ 1 ( S ) for a ll S ∈ dom τ 1 . Hence, there are fewer than κ -many poss ibilities for U 1 and τ 1 given U 0 and τ 0 .  An exceptional homoge ne o us compa c tum Y is co nstructed in [16] with X = T and w ( Y S ) = π ( Y S ) = c and χ ( Y S ) = ω for all S ∈ S . Hence, N t ( Y S ) ≤ c + and χN t ( Y S ) = ω for each S ∈ S . Moreover, ea ch Y S is 2 γ ordered lexic o graphica lly where γ is a ﬁxed indecomp osable ordinal in ω 1 \ ( ω + 1). Since cf γ = ω , it is ea sy to construct an ω op -like π -base of this spa ce. Hence, by Theorem 2.18, N t ( Y ) ≤ c + and π N t ( Y ) = χN t ( Y ) = ω . Thus, Obser v atio n 1.4 is justiﬁed for N t ( · ), π N t ( · ), and χN t ( · ). It remains to justify Observ ation 1.4 for χ K N t ( · ). W e ﬁrst no te that all known homogeneous compacta are contin uous images of pro ducts o f co mpacta each of weigh t a t most c . (Moreover, it it shown in [16] that an y Z as in Deﬁnition 2.17 is a contin uous image of X × Q S ∈ S Y S .) Therefore, the following theorem will suﬃce. Theorem 2.19. Supp ose Y is a c ontinu ous image of a pr o du ct X = Q i ∈ I X i of c omp acta. Then χ K N t ( Y ) ≤ sup i ∈ I w ( X i ) Before pr oving the a b ove theor em, we ﬁrst prove tw o lemmas. Deﬁnition 2.20. Given s ubsets P and Q o f a common p ose t, deﬁne P and Q to be mut u al ly dense if fo r all p 0 ∈ P a nd q 0 ∈ Q there exist p 1 ∈ P a nd q 1 ∈ Q such that p 0 ≥ q 1 and q 0 ≥ p 1 . Lemma 2. 2 1. L et κ b e a c ar dinal and let P and Q b e mutual ly dense subsets of a c ommon p oset. Then P is almost κ op -like if and only if Q is. Pr o of. Supp ose D is a κ op -like dense subset of P . Then it suﬃces to construc t a κ op -like dense subset of Q . Deﬁne a partial map f from | D | + to Q as follows. Se t f 0 = ∅ . Supp ose α < | D | + and we have constructed a pa rtial map f α from α to Q . Set E = { d ∈ D : d 6≥ q for all q ∈ ra n f α } . If E = ∅ , then set f α +1 = f α . Otherwise, choose q ∈ Q such that q ≤ e for s o me e ∈ E , and le t f α +1 be the smallest function extending f α such that f α +1 ( α ) = q . F or limit o r dinals γ ≤ | D | + , set f γ = S α<γ f α . Set f = f | D | + . Let us show that ra n f is κ op -like. Suppo se otherwise . Then there exists q ∈ ran f and an incr easing sequence h ξ α i α<κ in dom f s uch that q ≤ f ( ξ α ) for all α < κ . By the way we constr ucted f , there exists h d α i α<κ ∈ D κ such that f ( ξ β ) ≤ d β 6 = d α for a ll α < β < κ . Cho ose p ∈ P s uc h that p ≤ q . Then choo s e d ∈ D such that d ≤ p . Then d ≤ d β 6 = d α for all α < β < κ , which c o nt radicts that D is κ op -like. Therefore, ra n f is κ op -like. Finally , let us show that ran f is a dense s ubset of Q . Supp ose q ∈ Q . Cho o se p ∈ P suc h tha t p ≤ q . Then choose d ∈ D such that d ≤ p . By the wa y we constructed f , there exists r ∈ r an f such that r ≤ d ; hence, r ≤ q .  8 DA VID MILO VICH Lemma 2.22 . Supp ose f : X → Y is a c ontinuous su rje ction b etwe en c omp acta and C is close d in Y . Then χN t ( f − 1 C, X ) = χN t ( C, Y ) . Pr o of. Let A be a neigh b orho o d base o f C . By Lemma 2.21, it suﬃces to show that { f − 1 V : V ∈ A} is a neighborho o d base of f − 1 C . Supp ose U is a neighborho o d of f − 1 C . B y no rmality of Y , w e hav e f − 1 C = T V ∈A f − 1 V . By compa ctness of X , we hav e f − 1 V ⊆ U for some V ∈ A . Th us, { f − 1 V : V ∈ A} is a neighborho o d base of f − 1 C as desired.  Pr o of of The or em 2.19. By Lemma 2.2 2, we may assume Y = X . By Theor e m 2 .2, we may assume I is ﬁnite. Apply Theorem 2.5.  How sharp a re the b ounds o f Obse rv a tion 1.4? (3) is trivially sharp as e very s pa ce has lo cal No etherian type at least ω . W e will sho w that there is a homogeneous compactum w ith Noethia n t yp e c + , namely , the double arr ow space. Moreover, w e will show that Suslin lines hav e uncountable Noetheria n π -type . It is known to b e consistent that there are homogeneo us compact Suslin lines, but it is also known to be consis tent that there are no Suslin lines. It is no t clear whether it is consistent that all homo g eneous compa cta hav e No ether ian π -t yp e ω , even if we restrict to the ﬁrs t countable case. Also, it is not cle ar in any model of ZF C whether all ﬁrst countable homogeneous compacta have compact Noether ian type ω . Question 2.23 . Is there a ﬁrs t countable compactum with uncountable compact No etherian type? The following prop osition is essentially due to Peregudov [19]. Prop ositio n 2.2 4. If X is a sp ac e and π ( X ) < cf κ ≤ κ ≤ w ( X ) , then N t ( X ) > κ . Pr o of. Supp ose A is a bas e of X a nd B is π -base of X of size π ( X ). Then |A| ≥ κ ; hence, there exist U ∈ [ A ] κ and V ∈ B such that V ⊆ T U . Hence, there exists W ∈ A such that W ⊆ V ⊆ T U ; hence, A is not κ op -like.  Example 2.25. The double ar row space, deﬁned as ((0 , 1] × { 0 } ) ∪ ([0 , 1) × { 1 } ) ordered lexico graphically , has π - weigh t ω and weight c , a nd is is k nown to b e compact and homogeneous. By P rop osition 2.24, it has No ether ian type c + . Theorem 2.26. Supp ose X is a Suslin line. Then π N t ( X ) ≥ ω 1 . Pr o of. Let A b e a π -base of X consisting only of op en in terv als. By Lemma 2.21, it suﬃces to show that A is not ω op -like. Construct a sequence hB n i n<ω of max imal pairwise disjoint s ubsets of A as follows. Cho ose B 0 arbitrar ily . Given n < ω and B n , choo se B n +1 such that it r eﬁnes B n and B n ∩ B n +1 ⊆ [ X ] 1 . Let E denote the set of a ll endp oints o f interv als in S n<ω B n . Since X is Suslin, there exists U ∈ A \ [ X ] 1 such that U ∩ E = ∅ . F or each n < ω , the s et S B n is dense in X b y maximality; whence, there exists V n ∈ B n such that U ∩ V n 6 = ∅ . Since U ∩ E = ∅ , we hav e U ⊆ T n<ω V n . Thus, A is not ω op -like.  MA + ¬ CH implies there are no Souslin lines. It is not clear whether it further implies every homogeneo us compactum has No etherian π -type ω . Howev er, the next theor em gives us a pa rtial result. First, we need a lemma v ery similar to the result that MA + ¬ CH implies all Aro nsza jn trees ar e sp ecial. Deﬁnition 2.27. Given a subset E of a p os et Q , let ↑ Q E denote the set of q ∈ Q for which q has a low er b ound in E . NOETHERIAN TYPES OF HOMOGENEOUS COMP A CT A AND DY ADIC COMP ACT A 9 Lemma 2 .28. Assume MA. Su pp ose Q is an ω op 1 -like p oset of size less than c . Then Q is almost ω op -like or Q has an u nc ountable c enter e d su bset. Pr o of. Set P = [ Q ] <ω and or der P such that σ ≤ τ if a nd only if σ ∩ ↑ Q τ = τ . A suﬃciently g eneric ﬁlter G of P will b e such that S G is a dense ω op -like subset of Q . Hence, if P is ccc, then Q is a lmost ω op -like. Hence, we may assume P has an antic hain A of size ω 1 . W e may assume A is a ∆-system with ro ot ρ . Since Q is ω op 1 -like, we may a ssume σ ∩ ↑ Q ρ = ρ for all σ ∈ A . Cho o se a bijectio n h a α i α<ω 1 from ω 1 to A . W e may a ssume there exists an n < ω such that | a α \ ρ | = n fo r all α < ω 1 . F or each α < ω 1 , c ho ose a bijection h a α,i i i ω and the theorem is tr ue for all free b o olean algebras of size less than κ . W e will construct a contin uous elementary c hain h M α i α<κ of elementary sub- mo dels o f H θ and a contin uous increa sing sequence of sets h D α i α<κ satisfying the following co nditions for a ll α < κ . (1) α ∪ { B , ∧ , ∨ , Q } ⊆ M α and | M α | ≤ | α | + ω . (2) D α is a dense subset of Q ∩ M α . (3) D α ∩ ↑ q is ﬁnite for a ll q ∈ Q ∩ M α . (4) D α +1 ∩ ↑ q = D α ∩ ↑ q for all q ∈ Q ∩ M α . Given this constructio n, set D = S α<κ D α . Then D is a dense s ubs et of Q by (2 ). Moreov er, if α < κ and d ∈ D α , then d ∈ Q ∩ M α by (2); whence, d is b elow at most ﬁnitely many ele ments of D by (3) and (4 ). Hence , Q is almos t ω op -like. F or s tage 0, c ho ose any M 0 ≺ H θ satisfying (1). Since Q ∩ M 0 ⊆ B ∩ M 0 , we may choos e D 0 to be an ω op -like dense s ubset of Q ∩ M 0 , exactly what (2) a nd (3) r e q uire. At limit stages, (1) and (2) are clearly pr e served, and (3) is preserved bec ause of (4 ). 12 DA VID MILO VICH F or a s uccessor stage α + 1 , choo se M α +1 such that M α ≺ M α +1 ≺ H θ and (1 ) holds for stag e α + 1. Since Q ∩ M α +1 ⊆ B ∩ M α +1 , there is an ω op -like dense subset E of Q ∩ M α +1 . Set D α +1 = D α ∪ ( E \ ↑ ( Q ∩ M α )). Then (4) is easily veriﬁed: if q ∈ Q ∩ M α , then D α +1 ∩ ↑ q = ( D α ∩ ↑ q ) ∪ (( E ∩ ↑ q ) \ ↑ ( Q ∩ M α )) = D α ∩ ↑ q . Let us verify (2) for stage α + 1 . Let q ∈ Q ∩ M α +1 . If q ∈ ↑ ( Q ∩ M α ), then q ∈ ↑ D α ⊆ ↑ D α +1 bec ause o f (2) for stage α . Supp ose q 6∈ ↑ ( Q ∩ M α ). Cho os e e ∈ E such tha t e ≤ q . Then e 6∈ ↑ ( Q ∩ M α ); hence, q ∈ ↑ ( E \ ↑ ( Q ∩ M α )) ⊆ ↑ D α +1 . It r emains only to verify (3) for stage α + 1 . Let q ∈ Q ∩ M α +1 . T he n E ∩ ↑ q is ﬁnite; hence, by the deﬁnition of D α +1 , it suﬃces to show that D α ∩ ↑ q is ﬁnite. By Le mma 3.1, there ex ists r ∈ B ∩ M α such that r ≥ q and M α ∩ ↑ q = M α ∩ ↑ r ; hence, D α ∩ ↑ q = D α ∩ ↑ r . Since q ∈ Q , we hav e r ∈ M α ∩ ↑ Q . By elementarit y , there exists p ∈ Q ∩ M α such that p ≤ r ; hence, D α ∩ ↑ r ⊆ D α ∩ ↑ p . By (2) for stage α , w e hav e D α ∩ ↑ p is ﬁnite; he nc e , D α ∩ ↑ q is ﬁnite.  Deﬁnition 3.3. F o r any spac e X , let Clop( X ) denote the bo olean alg ebra o f clopen subsets of X . Theorem 3.4. L et X b e a dyadic c omp actum and let U b e a family of subsets of X su ch that for al l U ∈ U ther e exists V ∈ U such that V ∩ X \ U = ∅ . Then U is almost ω op -like. Pr o of. Let f : 2 κ → X b e a co n tinuous sur jection for some cardinal κ . Set B = Clop(2 κ ). Then B is a free b o olea n algebra . Set V = { f − 1 U : U ∈ U } . Then it suﬃces to show that V is almo st ω op -like. Let Q denote the set o f a ll B ∈ B such that V ⊆ B for s ome V ∈ V . By Theore m 3.2, Q is almost ω op -like. Hence, by Lemma 2.2 1, it suﬃces to show tha t Q and V a re mutually dens e . By deﬁnition, every Q ∈ Q contains s ome V ∈ V ; hence, it suﬃces to s how that ev ery V ∈ V contains some Q ∈ Q . Suppo se V ∈ V . Cho ose U ∈ U such that U ∩ X \ f [ V ] = ∅ . Then there exists B ∈ B such that f − 1 U ⊆ B ⊆ V ; hence, V ⊇ B ∈ Q .  The following corolla ry is immediate and it implies the ﬁrst half of Theorem 1.5. Corollary 3.5. L et X b e a dyadi c c omp actum . Th en, for al l close d subsets C of X , every neighb orho o d b ase of C c ontains an ω op -like neighb orho o d b ase of C . Mor e over, every π -b ase of X c ontains an ω op -like π -b ase of X . R emark. The ﬁrst half of the ab ove co rollar y can also prov ed s imply by citing Theorem 2.19 and Lemma 2 .21. Next we sta te the natural g eneralizatio ns of Lemma 3.1, Theo rem 3.2, Theo- rem 3.4, and Co r ollary 3.5 to contin uous imag es of pro ducts of compacta with bo unded weight . W e will only r emark brie ﬂy ab out the pr o ofs o f these genera liza- tions, for they ar e easy mo diﬁcations of the corr e spo nding o ld pro o fs. Lemma 3 . 6. L et κ b e a r e gular u nc ountable c ar dinal and let B b e a c opr o duct ` i ∈ I B i of b o ole an algebr as al l of size less than κ ; let { B , ∧ , ∨ , h B i i i ∈ I } ⊆ M ≺ H θ and M ∩ κ ∈ κ + 1 . Then, for al l q ∈ B , ther e exists r ∈ B ∩ M such that, for al l p ∈ B ∩ M , we have p ≥ q if and only if p ≥ r . In p articular, r ≥ q . Pr o of. Note that the subalgebra B ∩ M is the subco pro duct ` i ∈ I ∩ M B i naturally embedded in B . Then pro ceed a s in the pro of of Lemma 3 .1 with S i ∈ I B i , naturally embedded in B , playing the role of G .  NOETHERIAN TYPES OF HOMOGENEOUS COMP A CT A AND DY ADIC COMP ACT A 13 Theorem 3.7 . L et κ ≥ ω and B b e a c opr o duct of b o ole an algebr as al l of size at most κ . Then every subset of B is almost κ op -like. Pr o of. The pr o of is ess en tially the pro of o f Theorem 3 .2. Ins tead of using Lemma 3.1, use the instance of Lemma 3.6 for the regular uncountable cardinal κ + .  Theorem 3.8. L et κ ≥ ω and let X b e Hausdorﬀ and a c ontinuous image of a pr o duct of c omp acta al l of weight at most κ ; let U b e a family of subsets of X such that, for al l U ∈ U , ther e ex ist s V ∈ U such that V ∩ X \ U = ∅ . Then U is almost κ op -like. Pr o of. Let h : Q i ∈ I X i → X b e a con tinuous surjectio n where each X i is a com- pactum with weigh t at mos t κ . Ea ch X i embeds into [0 , 1] κ and is therefor e a contin uous imag e of a closed subspa ce o f 2 κ . Hence, we may a ssume Q i ∈ I X i is totally disconnected. The rest of the pro o f is just the pro of o f Theorem 3.4 with Theorem 3.7 replacing Theo rem 3.2.  The following corolla ry is immediate. Corollary 3.9. L et κ ≥ ω and let X b e Hausdorﬀ and a c ontinuous image of a pr o duct of c omp acta al l of weight at most κ . Then, for al l close d subsets C of X , every neighb orho o d b ase of C c ontains a κ op -like neighb orho o d b ase of C . Mor e over, every π -b ase of X c ontains a κ op -like π -b ase of X . R emark. Again, the ﬁrst half of the ab ov e corolla ry ca n also pro ved simply b y citing Theorem 2.19 and Lemma 2 .21. In co nt ras t to Corolla ry 3.5, not all dyadic compacta hav e ω op -like bases. The following pro p o sition is essentially due to Peregudov (see Lemma 1 of [19]). It makes it easy to pro duces exa mples of dyadic c o mpacta X such that N t ( X ) > ω . Prop ositio n 3.10. Su pp ose a p oint p in a sp ac e X satisﬁes π χ ( p, X ) < cf κ = κ ≤ χ ( p, X ) . Then N t ( X ) > κ . Pr o of. Let A b e a ba se of X . Let U 0 and V 0 be, resp ectively , a lo cal π -base at p of siz e at mo st π χ ( p, X ) and a lo ca l base at p of size χ ( p, X ). F or each element of U 0 , choo se a subset in A , ther eby pr o ducing a lo cal π -base U at p that is a subset of A of size at mos t π χ ( p, X ). Similarly , for each element of V 0 , cho ose a smaller neighborho o d of p in A , thereby producing a lo cal ba se V at p that is a subset o f A of size χ ( p, X ). Every elemen t of V c ontains an element of U . Hence, s ome element of U is co nt ained in κ -many elemen ts of V ; hence, A is not κ op -like.  Example 3.1 1. Let X b e the discr ete sum of 2 ω and 2 ω 1 . Let Y b e the quotient of X resulting from c ollapsing a p oint in 2 ω and a p oint in 2 ω 1 to a sing le p oint p . Then π χ ( p, Y ) = ω and χ ( p, Y ) = ω 1 ; hence, N t ( Y ) > ω 1 . Question 3.12 . Is there a dyadic compactum X such that π χ ( p, X ) = χ ( p, X ) for all p ∈ X but X has no ω op -like base? In pa r ticular, if Y is as in E xample 3.11 and Z is the discr ete sum o f Y and 2 ω 2 , then do es Z ω 1 hav e an ω op -like base? If we make an a dditio nal a ssumption ab out a dyadic c ompactum X , namely , that all its points hav e π - ch ara cter equal to its weigh t, then X has a n ω op -like base. Also, w e may choose this ω op -like bas e to b e a subs et of a n arbitrar y base of X . T o prov e this, w e approximate such an X by metric compacta. Each such metric compactum is constructed using the following technique due to Bandlow [2]. 14 DA VID MILO VICH Deﬁnition 3 . 13. Given a spac e X , let C ( X ) de no te the set of co nt inuous maps from X to R . Deﬁnition 3 . 14. Suppo se X is a space and F ⊆ C ( X ). F or all p ∈ X , let p/ F denote the set of q ∈ X satis fying f ( p ) = f ( q ) for all f ∈ F . F or each f ∈ F , deﬁne f / F : X/ F → R by ( f / F )( p/ F ) = f ( p ) for a ll p ∈ X . Lemma 3.15. Supp ose X is a c omp actum and F ⊆ C ( X ) . Then X/ F (with the quotient top olo gy) is a c omp actum and its top olo gy is the c o arsest t op olo gy for which f / F is c ontinuous for al l f ∈ F . F u rther su pp ose { X \ f − 1 { 0 } : f ∈ F } is a b ase of X and F ∈ M ≺ H θ . Then { ( X \ f − 1 { 0 } ) / ( F ∩ M ) : f ∈ F ∩ M } is a b ase of X/ ( F ∩ M ) . Pr o of. If f ∈ F , then f / F is clea rly contin uous with r esp ect to the quotient top ol- ogy of X/ F . Ther efore, the c ompact quotient top ology on X/ F is ﬁner than the Hausdorﬀ to po logy induced b y { f / F : f ∈ F } . If a compact top ology T 0 is ﬁner than a Hausdorﬀ to po logy T 1 , then T 0 = T 1 . Hence, the quotient top olog y on X/ F is the topo logy induced by { f / F : f ∈ F } . Set A = { X \ f − 1 { 0 } : f ∈ F } . Supp os e A is a base of X and F ∈ M ≺ H θ . Let us show that { ( X \ f − 1 { 0 } ) / ( F ∩ M ) : f ∈ F ∩ M } is a base of X/ ( F ∩ M ). Let U denote the set of pr eimages of op en ratio nal interv als with res pec t to elements of F ∩ M . Let V denote the set of none mpty ﬁnite intersections of elements of U . Then V ⊆ M a nd { V / ( F ∩ M ) : V ∈ V } is base o f X/ ( F ∩ M ). Supp ose p ∈ V 0 ∈ V . Then it suﬃces to ﬁnd W ∈ A ∩ M such that p ∈ W ⊆ V 0 . Cho ose V 1 ∈ V such that p ∈ V 1 ⊆ V 1 ⊆ V 0 . Then there exist n < ω a nd W 0 , . . . , W n − 1 ∈ A s uch that V 1 ⊆ S i | N 1 | . NOETHERIAN TYPES OF HOMOGENEOUS COMP A CT A AND DY ADIC COMP ACT A 15 (7) h Σ β i β ≤ α = Ψ( h M β i β <α ) . Mor e over, | Σ λ | = 1 and { α < λ : | Σ α | = 1 } is close d unb ounde d in λ for al l inﬁn ite c ar dinals λ ≤ η . Pr o of. Let Ω deno te the class of h γ i i i | γ j | for all i < j < n and | γ n − 1 | < κ if n > 0. Order Ω lexicog raphically and let Υ be the order iso mo rphism from O n to Ω. Given any σ = h γ i i i | N α,j | for all i < j < n . Let Υ( α i ) = φ i (Υ( α )) for all i < n . If i < j < n , then { N α,k : k < j } = Σ α j − 1 ; whence, either N α,j = ∅ or N α,i ∈ M α j − 1 ⊆ N α,j , dep ending on whether β j = 0. Thu s, (5 ) and (6) hold. Finally , let us pr ove (2). P ro ceed by induction o n α . Supp ose β n − 1 > 0 . Since { N α,i : i < n − 1 } = Σ α n − 1 and α n − 1 + β n − 1 = α , it suﬃces to show that | N α,n − 1 | ⊆ N α,n − 1 ≺ h H θ , . . . i . If β n − 1 ∈ Lim, then N α,n − 1 = S γ <β n − 1 N α n − 1 + γ ,n − 1 ; hence, | N α,n − 1 | ⊆ N α,n − 1 ≺ h H θ , . . . i . If β n − 1 6∈ Lim, then N α,n − 1 = N α − 1 ,n − 1 ∪ M α − 1 = M α − 1 bec ause N α − 1 ,n − 1 ∈ M α − 1 and | N α − 1 ,n − 1 | < κ ; hence, | N α,n − 1 | ⊆ N α,n − 1 ≺ h H θ , . . . i . Therefore, we may ass ume β n − 1 = 0. Hence, Σ α = { N α,i : i < n − 1 } ; hence, we may assume n > 1 . Since { N α,i : i < n − 2 } = Σ α n − 2 and α n − 2 < α , it suﬃces to show that | N α,n − 2 | ⊆ N α,n − 2 ≺ h H θ , . . . i . If β n − 2 = κ , then N α,n − 2 = S γ <κ N α n − 2 + γ ,n − 2 ; hence, | N α,n − 2 | ⊆ N α,n − 2 ≺ h H θ , . . . i . Hence, we may as sume β n − 2 > κ . Let Υ ( δ γ ) = h β 0 , . . . , β n − 3 , γ , 0 i for all γ ∈ [ κ, β n − 2 ). If β n − 2 ∈ Lim, then N α,n − 2 = S κ ≤ γ < β n − 2 N δ γ ,n − 2 ; hence, | N α,n − 2 | ⊆ N α,n − 2 ≺ h H θ , . . . i . Hence, we may let β n − 2 = ε + 1. Supp ose | ε | = κ . Then N α,n − 2 = N δ ε ,n − 2 ∪ S γ <κ M δ ε + γ . If γ < κ , then φ n − 1 (Υ( δ ε + γ )) = Υ( δ ε ); whence, δ ε and γ a re deﬁnable fr o m δ ε + γ and κ ; whenc e , γ ∪ S ρ<γ M δ ε + ρ ⊆ M δ ε + γ . Hence, | N δ ε ,n − 2 | = κ ⊆ S γ <κ M δ ε + γ ≺ h H θ , . . . i . Moreover, since N δ ε ,n − 2 ∈ M δ ε , we hav e N δ ε ,n − 2 ⊆ S γ <κ M δ ε + γ ; hence, | N α,n − 2 | = κ ⊆ N α,n − 2 ≺ h H θ , . . . i . Therefore, we may assume | ε | > κ . Le t Υ( ζ γ ) = h β 0 , . . . , β n − 3 , ε, κ + γ , 0 i for all γ < | ε | . Then N α,n − 2 = N δ ε ,n − 2 ∪ S γ < | ε | N ζ γ ,n − 1 . If γ < | ε | , then Υ( ζ γ )( n − 1) = κ + γ ; whence, γ ∈ M ζ γ ⊆ N ζ γ +1 ,n − 1 . Hence, | ε | ⊆ S γ < | ε | N ζ γ ,n − 1 ≺ h H θ , . . . i . Since | N δ ε ,n − 2 | = | ε | a nd N δ ε ,n − 2 ∈ M δ ε ⊆ N ζ 0 ,n − 1 , we hav e N δ ε ,n − 2 ⊆ S γ < | ε | N ζ γ ,n − 1 . Hence, | N α,n − 2 | = | ε | ⊆ N α,n − 2 ≺ h H θ , . . . i .  Prop ositio n 3. 1 8. If X has a network c onsisting of at most w ( X ) -many close d subsets (in p articular, if X is r e gular), then every b ase of X c ontains a b ase of size at most w ( X ) . 16 DA VID MILO VICH Pr o of. Let A b e a n ar bitrary base of X ; let B be a base of X of size at most w ( X ); let N a netw ork of X consisting of at most w ( X )-many closed subsets. Since X is w ( X ) + -compact, we may c ho ose, for each h N , B i ∈ N × B sa tisfying N ⊆ B , so me U N ,B ∈ [ A ] ≤ w ( X ) such that N ⊆ S U N ,B ⊆ B . Then S {U N ,B : N ∋ N ⊆ B ∈ B } is a ba se of X and in [ A ] ≤ w ( X ) .  Lemma 3.19 . L et X b e a dyadi c c omp actum such t hat π χ ( p, X ) = w ( X ) for al l p ∈ X . L et A b e a b ase of X c onsisting only of c ozer o sets. Then A c ontains an ω op -like b ase of X . Pr o of. Set κ = w ( X ); by Prop osition 3.1 8, we may a s sume |A| = κ . Choos e F ⊆ C ( X ) such that A = { X \ g − 1 { 0 } : g ∈ F } . Let h : 2 λ → X b e a contin uous surjection for some cardinal λ . Let B b e the free b o olean alg ebra Clop(2 λ ). B y Lemma 2.9, w e may ass ume κ > ω . Let h M α i α<κ be an ω 1 -approximation s equence in h H θ , ∈ , F , h i ; s et h Σ α i α ≤ κ = Ψ( h M α i α<κ ) as deﬁned in Le mma 3.1 7. F or each α < κ , s et A α = A ∩ M α and F α = F ∩ M α . F or every H ⊆ A α , let H / F α denote { U / F α : U ∈ H} . By L e mma 3.1 5, A α / F α is a base of X/ F α . Since X/ F α is a metric compactum, ther e e x ists W α ⊆ A α such that W α / F α is a base of X/ F α satisfying (2), (3), and (4) of Lemma 2 .9. B y (2) of Lemma 2.9, we may choose, for ea ch U ∈ W α , some E α,U ∈ B ∩ M α such that h − 1 U ⊆ E α,U ⊆ h − 1 V for all V ∈ W α satisfying U ⊆ V . Set G α = { E α,U : U ∈ W α } . Suppo se G α is not ω op -like. Then there ex is t U ∈ W α and h V n i n<ω ∈ W ω α such that E α,U ( E α,V n 6 = E α,V m for all m < n < ω . Set Γ = { W ∈ W α : U ( W } . By (2) o f Lemma 2.9, Γ is ﬁnite; hence, b y (4) of Lemma 2 .9, there exists n < ω such that { W ∈ W α : V n ( W } 6⊆ Γ. Hence, there exists W ∈ W α such that W strictly con tains V n but not U . Hence, b y (3) of Lemma 2.9, E α,V n ⊆ h − 1 W ; hence, h − 1 U ⊆ E α,U ( E α,V n ⊆ h − 1 W ; hence, U ( W , which is abs urd. There fore, G α is ω op -like. Let V α denote the set of V ∈ W α satisfying U 6⊆ V for all nonempty o pe n U ∈ S Σ α . Let us show tha t V α / F α is a base o f X/ F α . If V ∈ V α , then P ( V ) ∩ W α ⊆ V α ; hence, it suﬃces to show that V α cov ers X . Since | S Σ α | < κ , every po int of X has a neighborho o d in A that do es not c o nt ain any no nempt y op en subset o f X in S Σ α . By compactness, there is cov er of X b y ﬁnitely many such neighborho o ds, say , W 0 , . . . , W n − 1 . By elementarity , w e may a s sume W 0 , . . . , W n − 1 ∈ A α . Then { W i : i < n } has a r e ﬁning cover S ⊆ W α . Hence, S ⊆ V α ; hence, V α cov ers X as desired. Let U α denote the set of U ∈ V α such that U ⊆ V for some V ∈ V α . Then U α / F α is clearly a base of X/ F α . Set E α = { E α,U : U ∈ U α } . Then E α is ω op -like bec ause it is a subse t of G α . F or all I ⊆ P (2 κ ), set ↑I = { H ⊆ 2 κ : H ⊇ I for some I ∈ I } . F or all H ⊆ 2 κ , set ↑ H = ↑{ H } . Set U = S α<κ U α and C = B ∩ ↑ { h − 1 U : U ∈ U } . F or a ll α ≤ κ , set D α = S β <α E β . Then we claim the following for all α ≤ κ . (1) D α is a dense subset of C ∩ S Σ α . (2) D α ∩ ↑ H is ﬁnite for all H ∈ C ∩ S Σ α . (3) If α < κ , then D α +1 ∩ ↑ H = D α ∩ ↑ H for all H ∈ C ∩ S Σ α . W e prove this claim by induction. F or stage 0 , the claim is v acuous. F or limit stages, (1) is clea r ly preserved, and (2 ) is preserved beca us e of (3). Supp o se α < κ and (1) and (2) hold for stag e α . Then it suﬃces to prove (3) for s tage α and to prov e (1) and (2) for stage α + 1. NOETHERIAN TYPES OF HOMOGENEOUS COMP A CT A AND DY ADIC COMP ACT A 17 Let us verify (3). Seeking a contradiction, suppo se H ∈ C ∩ S Σ α and D α +1 ∩ ↑ H 6 = D α ∩ ↑ H . Then E α ∩ ↑ H 6 = ∅ ; hence, there exists U ∈ U α such that H ⊆ E α,U . By (1), there exist β < α and W ∈ U β such that E β ,W ⊆ H . By deﬁnition, there exists V ∈ V α such that U ⊆ V . Hence, h − 1 W ⊆ E β ,W ⊆ H ⊆ E α,U ⊆ h − 1 V ; hence, W ⊆ V . Since W ∈ M β ⊆ S Σ α and V ∈ V α , we hav e W 6⊆ V , which yields our desired co nt radictio n. Let us verify (1 ) for stage α + 1 . By (1) for stage α , we ha ve D α +1 = D α ∪ E α ⊆  C ∩ [ Σ α  ∪ ( C ∩ M α ) = C ∩ [ Σ α +1 , so we just need to show dens eness. Let H ∈ C ∩ S Σ α +1 . If H ∈ S Σ α , then H ∈ ↑D α , so we may a ssume H ∈ M α . By elemen tarity , there exists U 0 ∈ U α such that h − 1 U 0 ⊆ H . Cho ose U 1 ∈ U α such that U 1 ⊆ U 0 . Then E α,U 1 ⊆ h − 1 U 0 ; hence, E α,U 1 ⊆ H . Hence, H ∈ ↑D α +1 . T o complete the pro o f of the claim, let us verify (2) for stage α + 1. B y (1) for stage α + 1, it suﬃces to prov e D α +1 ∩ ↑ H is ﬁnite for all H ∈ D α +1 . By (3), if H ∈ D α , then D α +1 ∩ ↑ H = D α ∩ ↑ H , which is ﬁnite by (1) and (2 ) for stag e α . Hence, we may assume H ∈ E α . Since E α is ω op -like, it suﬃces to show that D α ∩ ↑ H is ﬁnite. Since D α ⊆ S Σ α , it suﬃce s to s how that D α ∩ N ∩ ↑ H is ﬁnite for all N ∈ Σ α . Let N ∈ Σ α . By Lemma 3.1, there ex ists G ∈ B ∩ N such that G ⊇ H and B ∩ N ∩ ↑ H = B ∩ N ∩ ↑ G ; henc e , D α ∩ N ∩ ↑ H = D α ∩ N ∩ ↑ G . Since G ⊇ H ∈ C , we hav e G ∈ C . By (2) for stag e α , the set D α ∩ N ∩ ↑ G is ﬁnite; hence, D α ∩ N ∩ ↑ H is ﬁnite. Since U ⊆ A , it suﬃces to prove that U is an ω op -like base of X . Supp ose p ∈ V ∈ A . Then ther e exists α < κ such that V ∈ A α . Hence, ther e exists U ∈ U α such that p/ F α ∈ U / F α ⊆ V / F α ; hence, p ∈ U ⊆ V . Thus, U is a bas e of X . Let us show that U is ω op -like. Supp ose not. Then there exists α < κ a nd U 0 ∈ U α such that there exis t inﬁnitely many V ∈ U such that U 0 ⊆ V . Cho os e U 1 ∈ U α such that U 1 ⊆ U 0 . Supp ose β < κ and U 0 ⊆ V ∈ U β . Then E α,U 1 ⊆ h − 1 U 0 ⊆ h − 1 V ⊆ E β ,V . By (1) and (2), D κ is ω op -like; hence, there a re only ﬁnitely man y pos sible v alues for E β ,V . Therefore, there exist h γ n i n<ω ∈ κ ω and h V n i n<ω ∈ Q n<ω U γ n such that V m 6 = V n and E γ m ,V m = E γ n ,V n for all m < n < ω . Suppo se that for some δ < κ we hav e γ n = δ for all n < ω . Let i < ω a nd set Γ = { W ∈ W δ : V i ( W } . By (2) and (4) of Lemma 2.9, there ex ists j < ω such that { W ∈ W δ : V j ( W } 6⊆ Γ. Hence , there exists W ∈ W δ such that W strictly contains V j but not V i . By (3) of L e mma 2.9, V j ⊆ W . Hence, h − 1 V i ⊆ E δ,V i = E δ,V j ⊆ h − 1 W . Hence, V i ⊆ W . Since W do es not str ictly co n tain V i , w e must hav e V i = V i = W . Hence, h − 1 V i = E δ,V i = E δ,V 0 . Since i was arbitrar y chosen, we have V m = V n = h [ E δ,V 0 ] for a ll m, n < ω , whic h is absurd. Therefore, our suppo sed δ do es not exist; hence, we may a ssume γ 0 < γ 1 . By deﬁnitio n, there exists W ∈ V γ 1 such that V 1 ⊆ W . Therefore , h − 1 V 0 ⊆ E γ 0 ,V 0 = E γ 1 ,V 1 ⊆ h − 1 W ; hence, V 0 ⊆ W . Since V 0 ∈ M γ 0 ⊆ S Σ γ 1 and W ∈ V γ 1 , we hav e V 0 6⊆ W , which is absurd. Therefore, U is ω op -like.  Let us show that w e may r emov e the requirement that the base A in Lemma 3.19 consist only of cozer o s ets. Prop ositio n 3 .20. If X is a sp ac e and A is a ( w ( X ) + ) op -like b ase of X , then |A| ≤ w ( X ) . 18 DA VID MILO VICH Pr o of. Seeking a co n tradictio n, suppo se |A| > w ( X ). Let B b e a base of X o f size w ( X ). Then every ele men t of A contains an element of B . Hence, some U ∈ B is contained in w ( X ) + -many elements of A . Clearly U contains some V ∈ A , so A is not ( w ( X ) + ) op -like.  R emark. The ab ove prop osition’s pro of can be tr ivially modiﬁed to s how tha t if A is a ( π ( X ) + ) op -like π -base of X , then |A| ≤ π ( X ). Likewise, if p ∈ X and A is a ( χ ( p, X ) + ) op -like loca l base a t p , then |A| ≤ χ ( p, X ). Lemma 3. 21. Supp ose X is a sp ac e with n o isolate d p oints and χ ( p, X ) = w ( X ) for al l p ∈ X . F urther supp ose κ = cf κ ≤ min { N t ( X ) , w ( X ) } and X has a network c onsist ing of at m ost w ( X ) -many κ -c omp act sets. Then every b ase of X c ontains an N t ( X ) op -like b ase of X . Pr o of. Set λ = N t ( X ) and µ = w ( X ). Let A be an arbitrary base o f X ; let B b e a λ op -like base of X ; let N be a netw ork of X consisting of at most µ -man y κ -compa c t sets. By P rop osition 3.20, |B| ≤ µ . Let hh N α , B α ii α<µ enum era te {h N , B i ∈ N × B : N ⊆ B } . Constr uc t a sequenc e hG α i α<µ as follows. Suppos e α < µ and hG β i β <α is a s equence of elements of [ B ] <κ . F or each p ∈ N α , w e ha ve χ ( p, X ) = µ ≥ κ = cf κ ; hence, we may choose U α,p ∈ B s uch that p ∈ U α,p 6∈ S β <α G β . Cho ose σ α ∈  N α  <κ such that N α ⊆ S p ∈ σ α U α,p . Set G α = { U α,p : p ∈ σ α } . F or ea ch α < µ , choose F α ∈ [ A ] <κ such that N α ⊆ S F α ⊆ B α and F α reﬁnes G α . Set F = S α<µ F α , which is easily seen to b e a base of X . Let us show that F is λ op -like. Suppose not. Then, since κ = cf κ ≤ λ , there e x ist V ∈ F and I ∈ [ µ ] λ and h W α i α ∈ I ∈ Q α ∈ I F α such that V ⊆ T α ∈ I W α . F or e ach α ∈ I , there is a super set o f W α in G α . By induction, G α ∩ G β = ∅ for all α < β < µ ; hence, V has λ -many sup ersets in the λ op -like base B , which is absur d, for V ha s a subset in B .  R emark. If X is regular and lo ca lly κ -compa ct and κ ≤ w ( X ), then it is ea sily seen that X has a netw ork consisting of at most w ( X )-many κ -compact close d sets. Theorem 3. 22. L et X b e a dyadi c c omp actum such that π χ ( p, X ) = w ( X ) for al l p ∈ X . Then every b ase A of X c ontains an ω op -like b ase of X . Pr o of. By Lemma 3 .1 9, N t ( X ) = ω . Since w ( X ) = π χ ( p, X ) ≤ χ ( p, X ) ≤ w ( X ) for a ll p ∈ X , we may apply L emma 3 .21 to get a subset of A tha t is an ω op -like base of X .  Finally , let us prove the s econd half of Theo rem 1.5. Corollary 3 . 23. L et X b e a homo gene ous dyadic c omp actum with b ase A . Then A c ontains an ω op -like b ase of X . Pr o of. Eﬁmov [5 ] and Gerlits [7 ] independently pr ov ed that the π - character of ev ery dyadic compactum is e q ual to its weigh t. Since X is homo geneous, π χ ( p, X ) = w ( X ) for all p ∈ X . Hence, A contains an ω op -like base of X b y Theorem 3.22.  Note that a compactum is dyadic if and only if it a continous image of a pro duct of second countable compacta. Let us prov e ge ne r alizations o f Theor em 3.22 a nd Corollar y 3 .23 abo ut c ontin uo us images of pro ducts of compacta with b ounded weigh t. NOETHERIAN TYPES OF HOMOGENEOUS COMP A CT A AND DY ADIC COMP ACT A 19 Lemma 3.24. Su pp ose κ = cf κ > ω and X is a sp ac e such that π χ ( p, X ) = w ( X ) ≥ κ for al l p ∈ X . F u rther supp ose X has a network c onsisting of at most w ( X ) -m any κ -c omp act close d set s . Then every b ase of X c ontains a w ( X ) op -like b ase of X . Pr o of. Set λ = w ( X ) and let A be a n a rbitrary base of X . By Pr op osition 3 .18, we may assume |A| = λ . Le t N b e a net work of X co ns isting of at mo st λ -many κ -compact sets. Let h M α i α<λ be a co n tinuous element ary chain such that A , N , M α ∈ M α +1 ≺ H θ and for all α < λ . W e may also require that M α ∩ κ ∈ κ > | M α | for all α < κ a nd | M α | = | κ + α | for all α ∈ λ \ κ . F or each α < λ , set A α = A ∩ M α . Set B = S α<λ A α +1 \ ↑A α , which is clearly λ op -like. Let us show that B is a base o f X . Suppo se p ∈ U ∈ A . Cho ose N ∈ N such that p ∈ N ⊆ U . Cho ose α < λ suc h that N , U ∈ A α +1 . F or each q ∈ N , cho ose V q ∈ A \ ↑A α such that q ∈ V q ⊆ U . Then there exis ts σ ∈ [ N ] <κ such that N ⊆ S q ∈ σ V q . By elementarit y , we may assume h V q i q ∈ σ ∈ M α +1 . C ho o se q ∈ σ such that p ∈ V q . Then V q ∈ B and p ∈ V q ⊆ U . Thu s, B is a base of X .  Theorem 3.2 5 . L et κ ≥ ω and let X b e Hausdorﬀ and a c ontinuous image of a pr o duct of c omp acta e ach with weight at most κ . Supp ose π χ ( p, X ) = w ( X ) for al l p ∈ X . Then every b ase of X c ontains a κ op -like b ase. Pr o of. Let h : Q i ∈ I X i → X b e a con tinuous surjectio n where each X i is a com- pactum with weigh t at mos t κ . Ea ch X i embeds into [0 , 1] κ and is therefor e a contin uous image of a closed subspace of 2 κ . Hence, we may assume Q i ∈ I X i is to- tally disconnected. Set λ = w ( X ); b y Lemmas 2.9 and 3.2 4, we ma y assume λ > κ . By Theor em 3.2 2, we may assume κ > ω . Inductively construct a κ + -approximation sequence h M α i α<λ in h H θ , ∈ , C ( X ) , h, h Clop( X i ) i i ∈ I i as follows. F or each α < λ , let h N α,β i β <κ be an ω 1 -approximation s equence in h H θ , ∈ , C ( X ) , h, κ, h Clop( X i ) i i ∈ I , h M β i β <α i . Set h Γ α,β i β ≤ κ = Ψ( h N α,β i β <κ ) as deﬁned in Lemma 3.17; let { M α } = Γ α,κ . Set h Σ α i α ≤ λ = Ψ( h M α i α<λ ). Set F = C ( X ) ∩ S Σ λ and A = { X \ f − 1 { 0 } : f ∈ F } . Then A is a base of X . By Lemma 3.2 1, it suﬃces to construct a subset o f A that is a κ op -like base of X . F or ea ch α < λ , set F α = F ∩ M α . Let V α denote the s et of V ∈ A ∩ M α satisfying U 6⊆ V fo r all nonempty op en U ∈ S Σ α . Arguing a s in the pro of Lemma 3.19, V α / F α is a base o f X/ F α . F or each β < κ , let V α,β denote the set of all V ∈ V α ∩ N α,β satisfying U 6⊆ V for a ll nonempt y op en U ∈ S Γ α,β . Let R α,β denote the set of h U, V i ∈ V 2 α,β for which U ⊆ V ; set U α,β = dom R α,β ; set U α = S β <κ U α,β . Let us show that U α / F α is also a base of X/ F α . Supp ose p ∈ V ∈ V α . Extend { V } to a ﬁnite sub cov er σ of V α such that p 6∈ S ( σ \ { V } ). Cho ose β < κ suc h that σ ∈ N α,β . F or each q ∈ X , ch o ose V q, 0 , V q, 1 ∈ A suc h that q ∈ V q, 0 and there ex ists W ∈ σ such that U 6⊆ V q, 0 ⊆ V q, 1 ⊆ W for all nonempty op en U ∈ S Σ α ∪ S Γ α,β . Cho ose τ ∈ [ X ] <ω such that X = S q ∈ τ V q, 0 . By elementarity , we may assume h V q,i i h q,i i∈ τ × 2 ∈ N α,β . Choos e q ∈ τ such that p ∈ V q, 0 . Then V q, 0 ∈ U α,β and p ∈ V q, 0 ⊆ V . Thus, U α / F α is a base of X/ F α . Set B = Clop  Q i ∈ I X i  . F or each h U 0 , U 1 i ∈ S β <κ R α,β , choo se E α ( U 0 , U 1 ) ∈ B ∩ M α such that h − 1 U 0 ⊆ E α ( U 0 , U 1 ) ⊆ h − 1 U 1 . Set E α,β = E α [ R α,β ]. Set E α = S β <κ E α,β . Let us show that E α is κ op -like. Suppose β , γ < κ and E α,β ∋ H ⊆ 20 DA VID MILO VICH K ∈ E α,γ . Then it suﬃces to s how that γ ≤ β . Seeking a contradiction, suppo se β < γ . There exist h U 0 , U 1 i ∈ R α,β and h V 0 , V 1 i ∈ R α,γ such that H = E α ( U 0 , U 1 ) and K = E α ( V 0 , V 1 ). Hence, S Γ α,γ ∋ U 0 ⊆ V 1 ∈ V α,γ , in contradiction with the deﬁnition o f V α,γ . Set U = S α<λ U α and C = B ∩ ↑ { h − 1 U : U ∈ U } . F or a ll α ≤ λ , set D α = S β <α E β . Then we claim the following for all α ≤ λ . (1) D α is a dense subset of C ∩ S Σ α . (2) |D α ∩ ↑ H | < κ fo r all H ∈ C ∩ S Σ α . (3) If α < λ , then D α +1 ∩ ↑ H = D α ∩ ↑ H for all H ∈ C ∩ S Σ α . W e prove this claim by induction. F or stage 0 , the claim is v acuous. F or limit stages, (1) is clea r ly preserved, and (2 ) is preserved beca us e of (3). Supp o se α < κ and (1) and (2) hold for stag e α . Then it suﬃces to prove (3) for s tage α and to prov e (1) and (2) for stage α + 1. Let us verify (3). Seeking a contradiction, suppo se H ∈ C ∩ S Σ α and D α +1 ∩ ↑ H 6 = D α ∩ ↑ H . Then E α ∩ ↑ H 6 = ∅ ; hence, ther e exists V ∈ U α such that H ⊆ h − 1 V . By (1 ), there exist β < α and U ∈ U β and K ∈ E β such that h − 1 U ⊆ K ⊆ H . Hence, U ⊆ V . Since U ∈ M β ⊆ S Σ α and V ∈ V α , we have U 6⊆ V , which yields our desired co nt radictio n. Let us verify (1 ) for stage α + 1 . By (1) for stage α , we ha ve D α +1 = D α ∪ E α ⊆  C ∩ [ Σ α  ∪ ( C ∩ M α ) = C ∩ [ Σ α +1 , so we just need to show dens eness. Let H ∈ C ∩ S Σ α +1 . If H ∈ S Σ α , then H ∈ ↑D α , so we may as sume H ∈ M α . By e le men tarity , there exists U ∈ U α such that h − 1 U ⊆ H . Cho o s e β < κ such that U ∈ U α,β ; choose V ∈ U α,β such that V ⊆ U . Then E α ( V , U ) ⊆ H ; hence, H ∈ ↑D α +1 . The pro of o f the claim is completed b y noting that (2) for sta ge α + 1 can b e verﬁed just as in the pro of of Lemma 3.19, except that Lemma 3.6 is used in place of Lemma 3.1. Just a s in the pro of of Le mma 3.19, U is a base o f X ; hence, it suﬃces to show that U is κ op -like. Suppo se γ < λ and δ < κ a nd U ∈ U γ ,δ and hh ζ α , η α ii α<κ ∈ ( λ × κ ) κ and h W α i α<κ ∈ Q α<κ U ζ α ,η α and U ⊆ T α<κ W α . Then it suﬃces to show that W α = W β for some α < β < κ . Choo s e V ∈ U γ ,δ such that V ⊆ U . F or ea ch α < κ , choo se V α ∈ V ζ α ,η α such that W α ⊆ V α ; set H α = E ζ α ( W α , V α ). Then E γ ( V , U ) ⊆ T α<κ H α . By (1) and (2), D λ is κ op -like; hence, ther e exis ts J ∈ [ κ ] ω 1 such that H α = H β for all α, β ∈ J ; hence, W α ⊆ V β for all α, β ∈ J . If α, β ∈ J and ζ α < ζ β , then S Σ ζ β ∋ W α ⊆ V β , in contradiction with V β ∈ V ζ β . Hence, ζ α = ζ β for all α, β ∈ J . If α, β ∈ J and η α < η β , then S Γ ζ β ,η β ∋ W α ⊆ V β , in contradiction with V β ∈ V ζ β ,η β . Hence, η α = η β for all α, β ∈ J . Hence, { W α : α ∈ J } ⊆ N ζ min J ,η min J ; hence, W α = W β for some α < β < κ .  Lemma 3.26 . L et κ b e an unc ountable r e gular c ar dinal; let X b e a c omp actum such that w ( X ) ≥ κ and X is a c ontinu ous image of a pr o duct of c omp acta e ach with weight less than κ . Then π ( X ) = w ( X ) . Pr o of. It suﬃces to prove that π ( X ) ≥ κ . Se e k ing a co n tradictio n, supp ose A is a π -bas e of X of s ize less than κ . Let h X i i i ∈ I be a sequence of compacta each with weigh t less than κ a nd let h be a co n tinuous surjection fro m Q i ∈ I X i to X . Cho ose M ≺ H θ such that A ∪ { C ( X ) , h, h C ( X i ) i i ∈ I } ⊆ M and | M | = |A| . Cho o s e p ∈ M ∩ Q i ∈ I X i and set Y = { q ∈ Q i ∈ I X i : p ↾ ( I \ M ) = q ↾ ( I \ M ) } . NOETHERIAN TYPES OF HOMOGENEOUS COMP A CT A AND DY ADIC COMP ACT A 21 Then it suﬃces to show that h [ Y ] = X , for that implies κ ≤ w ( X ) ≤ w ( Y ) < κ . Seeking a contradiction, suppos e h [ Y ] 6 = X . Then there exis ts U ∈ A such that U ∩ h [ Y ] = ∅ . By elementarit y , there exists σ ∈ [ I ∩ M ] <ω and h V i i i ∈ σ such that V i is a nonempty op en subse t of X i for all i ∈ σ , and T i ∈ σ π − 1 i V i ⊆ h − 1 U . Hence, Y ∩ T i ∈ σ π − 1 i V i 6 = ∅ , in contradiction with U ∩ h [ Y ] = ∅ .  Deﬁnition 3.27. Given any car dina l κ , set log κ = min { λ : 2 λ ≥ κ } . Lemma 3.28 . L et κ b e an unc ountable r e gular c ar dinal; let X b e a c omp actum such that w ( X ) ≥ κ and X is a c ontinu ous image of a pr o duct of c omp acta e ach with weight less than κ . Then π χ ( X ) = w ( X ) . Pr o of. Let h X i i i ∈ I be a sequence of compacta each with weigh t less than κ a nd let h b e a contin uous surjection from Q i ∈ I X i to X . F or any s pa ce Y , we hav e π ( Y ) = π χ ( Y ) d ( Y ). Hence, w ( X ) = π ( X ) = π χ ( X ) d ( X ) by Lemma 3.26; hence, we may assume d ( X ) = w ( X ). Arguing as in the pro of of Lemma 3.26, if A is a π -base o f X and A ∪ { C ( X ) , h, h C ( X i ) i i ∈ I } ⊆ M ≺ H θ , then X is a contin uous image of Q i ∈ I ∩ M X i ; hence, w e may as sume | I | = π ( X ). B y 5.5 of [10], d ( X ) ≤ d ( Q i ∈ I X i ) ≤ κ · log | I | . By 2.37 of [10], d ( Y ) ≤ π χ ( Y ) c ( Y ) for all T 3 non-discrete spaces Y . Since κ is a calib er of X i for all i ∈ I , it is a lso a calib er of X ; hence, | I | = π ( X ) = d ( X ) ≤ π χ ( X ) κ ; hence, lo g | I | ≤ κ · π χ ( X ). Therefo r e, w ( X ) = d ( X ) ≤ κ · π χ ( X ); hence, we may assume w ( X ) = κ . Let h U α i α<κ enum era te a base o f X . F or each α < κ , cho o se p α ∈ U α . Since d ( X ) = w ( X ) = κ , there is no α < κ such that { p β : β < α } is dense in X . Since κ is a ca liber of X , w e ma y c ho ose p ∈ X \ S α<κ { p β : β < α } . It suﬃces to sho w that π χ ( p, X ) = κ . Seeking a co ntradiction, supp ose π χ ( p, X ) < κ . Then there exists α < κ such that { U β : β < α } contains a lo cal π -base at p ; hence , p ∈ { p β : β < α } , in contradiction with how we c hose p .  Theorem 3.2 9 . L et h X i i i ∈ I b e a se quenc e of c omp acta; let X b e a homo gene ous c omp actum; let h : Q i ∈ I X i → X b e a c ontinuous surje ction. If ther e is a r e gular c ar dinal κ such that w ( X i ) < κ ≤ w ( X ) for al l i ∈ I , then every b ase of X c ontains a (sup i ∈ I w ( X i )) op -like b ase. Ot herwise, w ( X ) ≤ sup i ∈ I w ( X i ) and every b ase of X c ontains a ( w ( X ) + ) op -like b ase. Pr o of. The latter case is a trivial application of Pro po sition 3 .18. In the former case, Lemma 3.2 8 implies π χ ( p, X ) = w ( X ) for all p ∈ X ; a pply Theorem 3.25.  Every kno wn homogeneo us co mpactum is a c o nt inuous ima ge of a pr o duct of compacta each with weigh t a t most c ; hence, Theor em 3.29 provides a uniform justiﬁcation for our obser v ation that a ll known homogeneous compacta have Noe- therian type at most c + . Analogo us ly , since every known homogeneous compactum is such a contin uous image, it has c + among its calib er s ; hence, it has cellularity at most c . Let us now turn to the s pec tr um of No etherian t yp es of dyadic compacta and a pro of of Theo rem 1.6. Theorem 3.3 0 . Le t κ and λ b e inﬁnite c ar dinals such that λ < κ . L et X b e the discr ete su m of 2 κ and 2 λ . L et Y b e the qu otient sp ac e induc e d by c ol lapsing h 0 i α<κ and h 0 i α<λ to a single p oint p . If λ < cf κ , t hen N t ( Y ) = κ + . If λ ≥ cf κ , then N t ( Y ) = κ . 22 DA VID MILO VICH Pr o of. Clear ly χ ( p, Y ) = κ a nd π χ ( p, Y ) = λ . Hence, if λ < cf κ , then κ + ≤ N t ( Y ) ≤ w ( Y ) + = κ + by P rop osition 3.1 0. Supp ose λ ≥ cf κ . W e still have κ ≤ N t ( Y ) b y Prop osition 3 .10, so it suﬃces to co nstruct a κ op -like base of Y . Le t ∼ b e the equiv alence relation suc h that Y = X / ∼ . In building a ba se o f Y , we pro ceed in the canonica l wa y when aw ay from p : for each µ ∈ { κ, λ } , set A µ = {{ x ∈ 2 µ : η ⊆ x } / ∼ : η ∈ Fn( µ, 2) a nd η − 1 { 1 } 6 = ∅} . Cho ose f 0 : κ → cf κ such tha t for all α < cf κ the pr eimage f − 1 0 { α } is b ounded in κ . Deﬁne f : [ κ ] <ω → cf κ by f ( σ ) = f 0 (sup σ ) for all σ ∈ [ κ ] <ω . Cho o se g 0 : λ → cf κ suc h that for all α < c f κ the preimage g − 1 0 { α } is un b o unded in λ . Deﬁne g : [ λ ] <ω → cf κ by g ( σ ) = g 0 (sup σ ) for all σ ∈ [ λ ] <ω . Set A p = [ α< cf κ n  { x ∈ 2 κ : x [ σ ] = { 0 }} ∪ { x ∈ 2 λ : x [ τ ] = { 0 }}  / ∼ : h σ , τ i ∈ f − 1 { α } × g − 1 { α } o . Set A = A κ ∪ A λ ∪ A p . Let us show that A is a κ op -like base of Y . The only nontrivial as p ect o f showing that A is a ba se of Y is verifying tha t A p is a lo c al base at p . Supp ose U is an o pen neighborho o d of p . Then there exist σ ∈ [ κ ] <ω and τ ∈ [ λ ] <ω such that  { x ∈ 2 κ : x [ σ ] = { 0 }} ∪ { x ∈ 2 λ : x [ τ ] = { 0 }}  / ∼⊆ U. Cho ose α < λ s uc h that sup τ < α and g 0 ( α ) = f ( σ ). Set τ ′ = τ ∪ { α } and V =  { x ∈ 2 κ : x [ σ ] = { 0 }} ∪ { x ∈ 2 λ : x [ τ ′ ] = { 0 }}  / ∼ . Then V ⊆ U and V ∈ A p bec ause f ( σ ) = g ( τ ′ ). Thu s, A is a base o f Y . Let us show that A is κ op -like. Supp ose U, V ∈ A and U ⊆ V . If U ∈ A κ , then, ﬁxing U , ther e are only ﬁnitely p ossibilities fo r V in A κ ; the sa me is true if κ is replaced by λ or p . Hence, we may assume U ∈ A i and V ∈ A j for some { i, j } ∈ [ { κ, λ, p } ] 2 . Since no element of A p is a subse t of a n ele ment of A κ ∪ A λ , we have i 6 = p . Hence, there exists η ∈ Fn( i, 2) such that U = { x ∈ 2 i : η ⊆ x } / ∼ . Since S A κ ∩ S A λ = ∅ , we have j = p . Hence, there exist σ ∈ [ κ ] <ω and τ ∈ [ λ ] <ω such that V =  { x ∈ 2 κ : x [ σ ] = { 0 }} ∪ { x ∈ 2 λ : x [ τ ] = { 0 }}  / ∼ . If i = κ , then σ ⊆ η − 1 { 0 } ; hence, ﬁxing U , there are o nly ﬁnitely ma n y po s - sibilities for σ , and a t mo st λ -many p ossibilities for τ . If i = λ , then τ ⊆ η − 1 { 0 } ; hence, ﬁxing U , there are o nly ﬁnitely many p ossibilities for τ , and at most | sup f − 1 0 { g ( τ ) }| <ω -many p ossibilities for σ given τ . Thus, there ar e fewer than κ -many po ssibilities for V given U . Thus, A is κ op -like.  Corollary 3. 3 1. If κ is a c ar dinal of unc ountable c oﬁnality, t hen ther e is a total ly disc onne cte d dya dic c omp actum with No etherian typ e κ + . If κ is a singular c ar dinal, then ther e is a total ly disc onne ct e d dyadic c omp actum with No etherian typ e κ . Pr o of. F or the ﬁrst case, apply Theor em 3.30 with λ = ω . F or the seco nd cas e, apply Theor em 3.30 with λ = cf κ .  Combining the abov e cor ollary with the following theor em (and a trivial example like N t (2 ω ) = ω ) immediately proves Theorem 1.6. NOETHERIAN TYPES OF HOMOGENEOUS COMP A CT A AND DY ADIC COMP ACT A 23 Theorem 3. 32. L et X b e a dyadic c omp actum with b ase A c ons ist ing only of c ozer o set s . If N t ( X ) ≤ ω 1 , then A c ont ains an ω op -like b ase of X . Henc e, no dyadic c omp actum has No etherian typ e ω 1 . Pr o of. Let Q b e an ω op 1 -like ba se of X of size w ( X ). Imp ort all the notation from the pro of of Le mma 3.1 9 verbatim, except that we require h M α i α<κ to b e an ω 1 -approximation sequence in h H θ , ∈ , F , h, Qi . Then U is an ω op -like subset of A a s b efore. On the other hand, V α / F α is not necessa rily a base of X/ F α for a ll α < κ . Howev er, w e will show that U is still a ba s e of X . In doing so, we will rep eatedly use the fact that if U, Q ∈ M ≺ H θ and U is a nonempty op en subset of X , then all sup ersets o f U in Q are in M b eca use { V ∈ Q : U ⊆ V } is a co un table element of M . Suppo se q ∈ Q ∈ Q . Then it suﬃces to ﬁnd U ∈ U such that q ∈ U ⊆ Q . Let β b e the leas t α < κ such that there exis ts A ∈ A α satisfying q ∈ A ⊆ A ⊆ Q . Fix such an A ∈ A β . F or ea ch p ∈ A , choos e h A p , Q p i ∈ A × Q such that p ∈ A p ⊆ Q p ⊆ Q p ⊆ Q . Since M β ∋ A ⊆ Q ∈ Q , we hav e Q ∈ M β . Hence, by elementarit y , we may assume there exists σ ∈  A  <ω such that hh A p , Q p ii p ∈ σ ∈ M β and A ⊆ S p ∈ σ A p . Cho ose p ∈ σ such that q ∈ A p . Supp ose Q p 6∈ S Σ β . Then all nonempty o p en subsets of Q p are also not in S Σ β ; hence, there exist U ∈ U β and V ∈ V β such that q / F β ⊆ U ⊆ V ⊆ A p ⊆ Q . Therefore, we may assume Q p ∈ S Σ β . Cho ose α < β such that Q p ∈ M α . Then Q ∈ M α bec ause Q p ⊆ Q . Hence, there e x ists τ ∈ [ A α ] <ω such that Q p ⊆ S τ ⊆ S τ ⊆ Q . Cho ose W ∈ τ such that q ∈ W . The n q ∈ W ⊆ W ⊆ Q , in co n tradiction with the minimalit y o f β . Thus, U is a base of X .  Question 3.33 . If κ is a n sing ula r c ardinal with coﬁnality ω , then is there a dy adic compactum with No etherian t yp e κ + ? Is there a dyadic co mpactum with weakly inaccessible No etherian t yp e? W e note that the sp ectrum of No etherian types of a ll compacta is trivial. Theorem 3.34. L et κ b e a r e gular unc ount able c ar dinal. Then ther e exists a total ly disc onne cte d c omp actum X such that N t ( X ) = κ and X has a P κ -p oint. Pr o of. Let X be the closed subspa ce of 2 κ consisting o f all f ∈ 2 κ for which f ( α ) = 0 or f [ α ] = { 1 } for all o dd α < κ . Fir s t, let us show tha t X has a κ op -like base. F or each σ ∈ Fn( κ, 2 ), set U σ = { f ∈ X : f ⊇ σ } . Let E deno te the set of σ ∈ Fn( κ, 2) for which sup dom σ is e ven and U σ 6 = ∅ . Set A = { U σ : σ ∈ E } , which is clea rly a base of X . Let us show that A is κ op -like. Supp ose σ, τ ∈ E a nd U σ ⊆ U τ . If sup do m σ < sup dom τ , then for ea ch f ∈ U σ the seque nce ( f ↾ sup dom τ ) ∪ {h sup dom τ , 1 − τ (sup dom τ ) i} ∪ {h β , 0 i : sup dom τ < β < κ } is in U σ \ U τ , which is absurd. Hence, s up dom τ ≤ sup dom σ ; hence, there are few er than κ -many po ssibilities for τ given σ . Thus, A is κ op -like. Finally , it suﬃces to show that h 1 i α<κ is a P κ -p oint of X , for a P κ -p oint must hav e local No etherian type at least κ . F or each α < κ , set σ α = {h 2 α + 1 , 1 i} . Then { U σ α : α < κ } is a lo cal ba s e at h 1 i α<κ . Moreover, U σ α ) U σ β for a ll α < β < κ . Since κ is regula r , it follows that h 1 i α<κ is a P κ -p oint.  Corollary 3.35. Every inﬁnite c ar dinal is the No etherian typ e of some total ly disc onne cte d c omp actum. 24 DA VID MILO VICH Pr o of. By Lemma 2.9, all totally disco nnected metric compacta have No etheria n t yp e ω . By Theor em 3 .3 4, if κ is a regula r uncountable c a rdinal, then there is a totally disconnected compactum X with No etherian type κ . If κ is a singula r cardinal, then there is a totally disconnected dyadic compactum with No etherian t yp e κ by C o rollar y 3.3 1.  4. Reflecting cones Notice tha t the pro ofs of Theo rems 3.4 and 3 .22 only indirectly use the h yp othesis that X is dyadic: if X is mere ly a contin uous Haus do rﬀ image of the Stone s pace of a b o olean algebra s atisfying the c onclusion o f Lemma 3.1, then it is routine to chec k that the pro o fs o f Theor ems 3 .4 and 3 .22 are still v alid. This pro mpts the question of whether there are nondyadic compacta X for whic h these pro ofs apply . Equiv alen tly , is there a b o olean alg ebra B such that B is not a suba lg ebra of a free bo o lean algebr a but the co nclusion of Lemma 3 .1 holds for B ? W e show that the answer is no if | B | ≤ ω 1 , and present s ome partial re sults for larger B . Deﬁnition 4.1. Let B be a b o olean alg e bra. W e say that B r eﬂe cts c ones to A if A is a subalgebra o f B and, for all q ∈ B , there exists r ∈ A such that for all p ∈ A we have p ≥ q if and only if p ≥ r . Let h H θ , . . . i denote an expansion of the {∈} -structure H θ to some L -structure for some countable lang uage L . W e say that B re ﬂects co nes in h H θ , . . . i if B reﬂects c ones to B ∩ M for all M ≺ h H θ , . . . i . If n < ω , then we s ay that B n -reﬂects cones in h H θ , . . . i if, for all Σ of size at most n satisfying (2), (5), and (6) of Lemma 3.17, B reﬂects cones to the subalgebra genera ted b y B ∩ S Σ. W e say that B ω -r e ﬂects cones in h H θ , . . . i if B n -reﬂects cones in h H θ , . . . i for a ll n < ω . W e say that B reﬂects cone s ( n -r e ﬂects cones, ω - r eﬂects cones) if B reﬂects cones ( n -reﬂects cones, ω -reﬂects cones) in some h H θ , . . . i for all suﬃciently large θ . Note that 1-reﬂe c ting cones is eq uiv alent to reﬂecting cones. Lemma 4.2. Supp ose B is b o ole an algebr a with element c and su b algebr a A such that B r eﬂe cts c ones to A . L et C denote the sub algebr a of B gener ate d by A ∪ { c } . Then B r eﬂ e cts c ones to C . Pr o of. Let q ∈ B . By hypothesis, ther e exist r 0 , r 1 ∈ A such tha t fo r a ll p ∈ A we hav e p ≥ q ∧ c if and only if p ≥ r 0 and p ≥ q ∧ c ′ if and only if p ≥ r 1 . In particular, r 0 ≥ q ∧ c a nd r 1 ≥ q ∧ c ′ . Set r = ( r 0 ∨ c ′ ) ∧ ( r 1 ∨ c ) ∈ C . Then r ≥ (( q ∧ c ) ∨ c ′ ) ∧ (( q ∧ c ′ ) ∨ c ) = q . Supp ose p ∈ C and p ≥ q . Then it suﬃces to show that p ≥ r . There exist p 0 , p 1 ∈ A such that p = ( p 0 ∨ c ) ∧ ( p 1 ∨ c ′ ). Then p 0 ∨ c ≥ q ; hence, p 0 ≥ q ∧ c ′ ; hence, p 0 ≥ r 1 ; hence, p 0 ∨ c ≥ r 1 ∨ c . B y symmetry , p 1 ∨ c ′ ≥ r 0 ∨ c ′ . Hence, p ≥ r .  Lemma 4.3. Supp ose B is b o ole an algebr a with element c and su b algebr a A such that B r eﬂe ct s c ones to A . F urt her su pp ose f is an emb e dding of A into a fr e e b o ole an algebr a. L et C denote the sub algebr a of B gener ate d by A ∪ { c } . Then f extends to an emb e dding of C into a fr e e b o ole an algebr a. Pr o of. W e may as sume f embeds A int o a free b o olea n alg ebra D s uch that ther e exists non trivial d ∈ D indep endent from the subalgebra f [ A ]. By h yp o thesis, there exist a 0 , a 1 ∈ A such that for all p ∈ A we hav e p ≥ c if and only if p ≥ a 0 and p ≥ c ′ if and o nly if p ≥ a 1 . Hence, for all p ∈ A , we hav e p ≤ c if and only if NOETHERIAN TYPES OF HOMOGENEOUS COMP A CT A AND DY ADIC COMP ACT A 25 p ≤ a ′ 1 . In particular, a 0 ≥ c ≥ a ′ 1 . Let g b e the unique homomo r phism from C to D ex tending f such tha t g ( c ) = ( f ( a 0 ) ∧ d ) ∨ f ( a 1 ) ′ . Suppose x, y ∈ C . Then it suﬃces to s how tha t x ≥ y if and only if g ( x ) ≥ g ( y ). There exist x 0 , x 1 , y 0 , y 1 ∈ A such that x = ( x 0 ∧ c ) ∨ ( x 1 ∧ c ′ ) and y = ( y 0 ∧ c ) ∨ ( y 1 ∧ c ′ ). Then x ≥ y if and only if x 0 ∧ c ≥ y 0 ∧ c and x 1 ∧ c ′ ≥ y 1 ∧ c ′ ; likewise, g ( x ) ≥ g ( y ) if and only if f ( x 0 ) ∧ g ( c ) ≥ f ( y 0 ) ∧ g ( c ) and f ( x 1 ) ∧ g ( c ) ′ ≥ f ( y 1 ) ∧ g ( c ) ′ . By symmetry , it suﬃces to show that x 0 ∧ c ≥ y 0 ∧ c if a nd only if f ( x 0 ) ∧ g ( c ) ≥ f ( y 0 ) ∧ g ( c ). Clearly x 0 ∧ c ≥ y 0 ∧ c if and only if x 0 ∨ y ′ 0 ≥ c ; likewise, f ( x 0 ) ∧ g ( c ) ≥ f ( y 0 ) ∧ g ( c ) if and o nly if f ( x 0 ) ∨ f ( y 0 ) ′ ≥ g ( c ). By deﬁnition, f ( x 0 ) ∨ f ( y 0 ) ′ ≥ g ( c ) if and only if f ( x 0 ) ∨ f ( y 0 ) ′ ≥ ( f ( a 0 ) ∧ d ) ∨ f ( a 1 ) ′ , which is eq uiv alent to f ( x 0 ) ∨ f ( y 0 ) ′ ≥ f ( a 0 ) bec ause f ( a 0 ) ≥ f ( a 1 ) ′ and d is indep endent from f [ A ]. Hence, x 0 ∧ c ≥ y 0 ∧ c if and o nly if x 0 ∨ y ′ 0 ≥ c if a nd only if x 0 ∨ y ′ 0 ≥ a 0 if and o nly if f ( x 0 ) ∨ f ( y 0 ) ′ ≥ f ( a 0 ) if and o nly if f ( x 0 ) ∨ f ( y 0 ) ′ ≥ g ( c ) if and o nly if f ( x 0 ) ∧ g ( c ) ≥ f ( y 0 ) ∧ g ( c ).  Theorem 4.4. Supp ose B is a b o ole an algebr a that ω - r eﬂ e cts c ones. Then B is a sub algebr a of a fr e e b o ole an algebr a. Pr o of. Let h M α i α< | B | be an ω 1 -approximation s equence in h H θ , ∈ , B , . . . i where B ω -reﬂec ts co nes in h H θ , ∈ , B , . . . i . Let h Σ α i α ≤| B | be as in Lemma 3 .17. F or each α ≤ | B | , let B α be the subalgebr a of B genera ted by B ∩ S Σ α . T r ivially , we may choose an em b edding f 0 of B 0 int o a fre e b o olea n algebra. Supp ose α < | B | a nd we hav e an embedding f α of B α int o a fr e e b o olea n alg ebra. Let B ∩ M α = { b n : n < ω } . F or each n < ω , let A n be the subalgebr a o f B gener ated by B α ∪ { b m : m < n } ; by r epea ted a pplication of Lemma 4.2, B r eﬂects cone s to each A n . Se t g 0 = f α . Suppo se n < ω and g n is a n embedding o f A n int o a free bo o lean a lgebra. By Lemma 4.3, there is an extensio n g n +1 of g n embedding A n +1 int o a free bo olean algebra. Set f α +1 = S n<ω g n . Then f α +1 is an e xtension of f α embedding B α +1 int o a free b o olean algebr a. F or limit ordinals α ≤ | B | , set f α = S β <α f β . Then f | B | embeds B into a fr e e b o olean alg ebra.  Corollary 4.5. L et n < ω and B b e a b o ole an algebr a of size at most ω n . If B n -r eﬂe cts c ones, then B is a sub algebr a of a fr e e b o ole an algebr a. Pr o of. If Σ satisﬁes (2), (5), and (6) of Lemma 3 .17 and | Σ | > n , then there exists N ∈ Σ s uc h that | B | ⊆ N . W e may assume B ∈ N ; hence, B ⊆ N ; hence, B ∩ S Σ = B . Thus, B ω -r eﬂects cones.  Question 4.6 . Is there a bo o lean algebr a that re ﬂects cones but is no t a s ubalgebra of a free b o olean algebra? 5. More o n l ocal Noetherian type In this section, we ﬁnd tw o suﬃcient conditio ns for a compactum to hav e a p oint with an ω op -like lo cal base. The ﬁrst of these conditions will be used to prov e Theorem 1.7. W e als o present some r elated results abo ut lo cal bases in terms of T uk ey re ducibility . Deﬁnition 5.1. Given cardinals λ ≥ κ ≥ ω and a subset E in a space X , a lo c al h λ, κ i - splitter at E is a set U of λ -many op en neighborho o ds of E such that E is contained in the in terior of T V for any V ∈ [ U ] κ . If p ∈ X , then w e call a loca l h λ, κ i -splitter at { p } a loc al h λ, κ i -splitter a p . 26 DA VID MILO VICH Theorem 5. 2. Supp ose X is a c omp actum and ω 1 ≤ κ = min p ∈ X π χ ( p, X ) . Then ther e is a lo c al h κ, ω i -splitter at some p ∈ X . Pr o of. Given a ny map f , let Q f denote {h x i i i ∈ dom f : ∀ i ∈ dom f x i ∈ f ( i ) } . Given any inﬁnite open f amily E , let Φ( E ) denote the set o f h σ, Γ i ∈ [ E ] <ω × ([ E ] ω ) <ω for which every τ ∈ Q Γ satisﬁes T σ ⊆ S ran τ . Then Φ( E ) = ∅ alwa ys implies E is ω op -like and centered. Let R denote the set of nonempty regula r op en subsets of X . Cho ose h W n i n<ω ∈ R ω such that W n +1 ( W n 6 = X for all n < ω . Let Ω denote the class of trans ﬁnite sequeces hh U α , V α ii α<η of elements of R 2 satisfying the following. (1) η ≥ ω and hh U n , V n ii n<ω = hh W n +1 , W n ii n<ω . (2) U α ⊆ V α for all α < η . (3) P ( V α ) ∩ n T σ \ S τ : σ, τ ∈  S β <α { U β , V β }  <ω o ⊆ {∅} for all α < η . (4) Φ  S α<η { U α , V α }  = ∅ . Seeking a contradiction, supp ose η is a limit ordinal and hh U α , V α ii α<η 6∈ Ω, but hh U β , V β ii β <α ∈ Ω for all α < η . Then (1), (2), and (3) hold for hh U α , V α ii α<η , so there exists h σ , Γ i ∈ Φ  S α<η { U α , V α }  . W e may choose i ∈ dom Γ s uc h that Γ( i ) 6⊆ S β <α { U β , V β } for all α < η . Set Λ = Γ ↾ (dom Γ \ { i } ). W e may assume dom Γ is minimal a mong its p ossible v alues; hence, there exists τ ∈ Q Λ such that T σ 6⊆ S ran τ . Cho o se α < η a nd W ∈ Γ( i ) s uch that σ ∪ ran τ ⊆ S β <α { U β , V β } and W ∈ { U α , V α } . Then T σ \ S ran τ 6⊆ W by (2) and (3). Since W is reg ular, T σ \ S ran τ 6⊆ W ; hence, T σ 6⊆ W ∪ S ran τ , in contradiction with h σ, Γ i ∈ Φ  S α<η { U α , V α }  . Thus, Ω is clo sed with r esp ect to unions o f increasing chains. It follows from (3) that Ω ⊆ ( R 2 ) < |R| + . Mo reov er, hh W n +1 , W n ii n<ω ∈ Ω. Hence, b y Z orn’s Lemma, Ω has a maximal element hh U α , V α ii α<η . Set B = S α<η { U α , V α } . Let us show tha t η ≥ κ . Supp ose not. F o r ea ch x ∈ X , cho ose Y x , Z x ∈ R such that x ∈ Y x ⊆ Y x ⊆ Z x and Z x do es not contain any nonempty op en set of the form T σ \ S τ where σ, τ ∈ [ B ] <ω . Cho ose ρ ∈ [ X ] <ω such that S x ∈ ρ Y x = X . Let us show that Φ ( B ∪ { Y x , Z x } ) = ∅ for some x ∈ ρ . Se e k - ing a co nt radictio n, supp o se h σ x , Γ x i ∈ Φ( B ∪ { Y x , Z x } ) for all x ∈ ρ . W e may assume S x ∈ ρ S ran Γ x ⊆ B . Let Λ b e a co ncatenation of { Γ x : x ∈ ρ } and set τ = B ∩ S x ∈ ρ σ i . Then for all ζ ∈ Q Λ we hav e \ τ = \ y ∈ ρ \ ( σ y ∩ B ) = [ x ∈ ρ Y x ∩ \ y ∈ ρ \ ( σ y ∩ B ) ! ⊆ [ x ∈ ρ \ σ x ⊆ [ ran ζ . Hence, h τ , Λ i ∈ Φ( B ), in contradiction with (4). Therefore, we may choose x ∈ ρ such that Φ( B ∪ { Y x , Z x } ) = ∅ . But then hh U α , V α ii α<η +1 ∈ Ω if we set U η = Y x and V η = Z x , in co nt radic tio n with the maximality o f hh U α , V α ii α<η . Thus, η ≥ κ . Set A = { V α : α < η } . By (3), |A| = | η | ≥ κ . Set K = T α<η U α . Then it suﬃces to show that A is a lo ca l h| η | , ω i -splitter at some x ∈ K . Suppos e not. Then ea ch x ∈ K has an o pen neighbo rho o d W x that is a subset o f inﬁnitely many elements of A . Hence, Φ( B ∪ { W x } ) 6 = ∅ for all x ∈ K . C ho o se ρ ∈ [ K ] <ω such that K ⊆ S x ∈ ρ W x . Choose a n op en set W suc h that W ∪ S x ∈ ρ W x = X and W ∩ K = ∅ . By co mpa ctness, B ∪ { W } is not centered; hence, Φ( B ∪ { W } ) 6 = ∅ . Reusing o ur ear lier concatena tion arg ument, we have Φ( B ) 6 = ∅ , in co ntradiction with (4). Thus, A is a lo ca l h| η | , ω i -splitter at so me x ∈ K .  NOETHERIAN TYPES OF HOMOGENEOUS COMP A CT A AND DY ADIC COMP ACT A 27 Lemma 5.3. Supp ose E is a subset of a sp ac e X and E has no ﬁn ite neigh- b orho o d b ase. Then χN t ( E , X ) is the le ast κ ≥ ω for which t her e is a lo c al h χ ( p, X ) , κ i -splitter at E . Pr o of. Set κ = χN t ( E , X ) and λ = χ ( E , X ). By Lemma 2 .4, λ ≥ κ ; hence, a κ op -like neigh b orho o d base of E (which necessarily ha s size λ ) is a lo cal h λ, κ i -splitter at E . T o s how the co nverse, let h U α i α<λ be a s equence of op en neighbor ho o ds of E . Let { V α : α < λ } be a neighborho o d base of E . F or each α < λ , choos e W α ∈ { V β : β < λ } such tha t W α ⊆ U α ∩ V α . Then { W α : α < λ } is a neigh- bo rho o d base of E . Let µ < κ . Then there e x ist α < λ and I ∈ [ λ ] µ such that W α ⊆ T β ∈ I W β . Hence, E is cont ained in the interior of T β ∈ I U β . Hence, { U α : α < λ } is not a lo cal h λ, µ i -splitter a t E .  Pr o of of The or em 2.14. W e may assume χ ( X ) ≥ ω 1 . By Theo rem 5.2, there is a lo cal h χ ( X ) , ω i -splitter at some p ∈ X . By Lemma 5 .3, χN t ( p, X ) = ω .  Pr o of of The or em 1.7. Let X b e a homo g eneous compactum. By a res ult of Ar han- gel ′ ski ˘ ı (see 1.5 of [1]), | Y | ≤ 2 π χ ( Y ) c ( Y ) for all homogeneous spa ces Y . Since | X | = 2 χ ( X ) by Arha ngel ′ ski ˘ ı’s Theorem and t he ˇ Cech-P ospi ˇ sil Theorem, we ha ve χ ( X ) ≤ π χ ( X ) c ( X ) by GCH. If π χ ( X ) = χ ( X ), then χ N t ( X ) = ω by Theo rem 2.1 4. Hence, we may a ssume π χ ( X ) < χ ( X ); hence, χN t ( X ) ≤ χ ( X ) ≤ c ( X ) by Theorem 2.5.  Example 5. 4 . Consider 2 ω 1 ordered lexicogr a phically . Every p oint in this spa c e has character a nd lo cal No etheria n type ω 1 , and some but not all p oints hav e π -character ω . Deﬁnition 5.5 (T uk ey [22]) . Given tw o qua siorders P and Q , we sa y f is a T u key map from P to Q and write f : P ≤ T Q if f is a map from P to Q such that all preimages of b ounded subsets of Q ar e b ounded in P . W e s ay that P is T u key r e ducible to Q and write P ≤ T Q if there exists f : P ≤ T Q . W e say that P a nd Q are T ukey e quivalent and write P ≡ T Q if P ≤ T Q ≤ T P . T uk ey showed that tw o directed sets are T ukey equiv ale nt if and only if they embed a s coﬁnal subsets of a c o mmon dir ected se t. In particular , a n y tw o lo cal ba ses at a common p oint in a topo lo gical space are T uk ey equiv a lent. Another, eas ily chec k ed fact is thats P ≤ T [cf P ] <ω for every dir ected set P . Also , [ κ ] <ω ≤ T [ λ ] <ω if κ ≤ λ . Lemma 5 . 6. Supp ose κ ≥ ω and E is a subset of a s p ac e X with a lo c al h κ, ω i -splitter at E . Then h [ κ ] <ω , ⊆i ≤ T hA , ⊇i for every neighb orho o d b ase A of E . Pr o of. Let U b e a lo cal h κ, ω i -splitter at E . Let N b e the set of open neigh bo rho o ds of E . Then N is T ukey equiv alent to every neighborho o d base of E (with resp ect to ⊇ ), so it suﬃces to show that [ U ] <ω ≤ T hN , ⊇i . Deﬁne f : [ U ] <ω → N by f ( σ ) = T σ for all σ ∈ [ U ] <ω . Then, for all N ∈ N , we hav e | f − 1 ↑ N | < ω bec ause U is a lo ca l h κ, ω i -splitter; whence, f − 1 ↑ N is bo unded in [ U ] <ω . Th us, f : [ U ] <ω ≤ T hN , ⊇i .  Theorem 5.7. S u pp ose X is a c omp actum and ω 1 ≤ κ = min p ∈ X π χ ( p, X ) . Then, for some p ∈ X , every lo c al b ase A at p satisﬁes h [ κ ] <ω , ⊆i ≤ T hA , ⊇i . Pr o of. Combine Theorem 5.2 a nd Lemma 5.6.  28 DA VID MILO VICH Lemma 5.8. Supp ose E is a subset of a sp ac e X and E has no ﬁnite neighb orho o d b ase. Then the fol lowing ar e e quivalent. (1) χN t ( E , X ) = ω . (2) Ther e is a lo c al h χ ( E , X ) , ω i -splitter at E . (3) Every neighb orho o d b ase A of E satisﬁes h [ χ ( E , X )] <ω , ⊆i ≡ T hA , ⊇i . Pr o of. By Lemma 5.3, (1) and (2) are equiv alent. Let B b e a neighbor ho o d base of E of size χ ( E , X ). By Lemma 5 .6, (2) implies [ χ ( E , X )] <ω ≤ T hA , ⊇i ≡ T hB , ⊇i ≤ T [ χ ( E , X )] <ω for every neighbor ho o d base A of E . Thus, (2) implies (3). Finally , supp ose A is a neighborho o d base of E and [ χ ( E , X )] <ω ≡ T hA , ⊇i . Then [ χ ( E , X )] <ω and hA , ⊇i embed as coﬁnal subsets of a co mmon directed se t. Hence, hA , ⊆i is almost ω op -like by Lemma 2.21. Hence, A contains an ω op -like neighborho o d base of E . Thus, (3) implies (1).  Theorem 5.9. Su pp ose X is an inﬁnite homo gene ous c omp actum and π χ ( X ) = χ ( X ) . Then, for al l p ∈ X and for al l lo c al b ases A at p , we have hA , ⊇i ≡ T h [ χ ( X )] <ω , ⊆i . Pr o of. Combine Theorem 2.1 4 and Lemma 5.8.  Deﬁnition 5 .10. Given n < ω and ordinals α, β 0 , . . . , β n , let α → ( β 0 , . . . , β n ) denote the prop os ition that for all f : [ α ] 2 → n + 1 there exist i ≤ n and H ⊆ α such that f [[ H ] 2 ] = { i } and H has o rder type β i . Lemma 5.11 . S u pp ose κ = cf κ > ω and P is a dir e cte d s et such t hat [ κ ] <ω ≤ T P . Then P c ontains a set of κ -many p airwise inc omp ar able elements. Pr o of. Let Q b e a well-founded, coﬁnal subs e t of P . Then P ≡ T Q ; let f : [ κ ] <ω ≤ T Q . Deﬁne g : [ κ ] 2 → 3 b y g ( { α < β } ) = 0 if f ( { α } ) 6≤ f ( { β } ) 6≤ f ( { α } ) and g ( { α < β } ) = 1 if f ( { α } ) > f ( { β } ) and g ( { α < β } ) = 2 if f ( { α } ) ≤ f ( { β } ). By the Erd¨ os-Dushnik-Miller Theorem, κ → ( κ, ω + 1 , ω + 1). Since Q is well-founded, there is no H ∈ [ κ ] ω such that g [[ H ] 2 ] = { 1 } . Since f is T uk ey a nd all inﬁnite subsets of [ κ ] <ω are un b ounded, there is no H ⊆ κ of or der type ω + 1 such that g [[ H ] 2 ] = { 2 } . Hence, there exists H ∈ [ κ ] κ such that g [[ H ] 2 ] = { 0 } ; whence, f [[ H ] 1 ] is a κ - sized, pairwis e inco mparable subs e t of P .  Theorem 5.12 . S upp ose κ = cf κ > ω and X is a c omp actum such t hat every p oint has a lo c al b ase not c ontaining a set of κ -many p airwise inc omp ar able elements. Then some p oint in X has π -char acter less than κ . Pr o of. Combine Theor em 5.7 and Lemma 5.1 1 to prov e the contrapos itive of the theorem.  Corollary 5. 1 3. Supp ose X is a c omp actum such t hat every p oint has a lo c al b ase that is wel l quasi-or der e d with re sp e ct to ⊇ . Then some p oint in X has c oun t able π -char acter. Finally , let us present a few r esults ab out lo cal No e therian t yp e and top olog ical embeddings. Lemma 5.14. Supp ose X is a sp ac e, Y ⊆ X , and p ∈ Y satisﬁes χ ( p, Y ) = χ ( p, X ) . Then χN t ( p, X ) ≤ χN t ( p, Y ) . NOETHERIAN TYPES OF HOMOGENEOUS COMP A CT A AND DY ADIC COMP ACT A 29 Pr o of. Set λ = χ ( p, Y ) a nd κ = χN t ( p, Y ); w e may ass ume λ > ω by Theorem 2.5. By Lemma 5.3, we may choo s e a loc a l h λ, κ i -splitter A a t p in Y . F or each U ∈ A , choose an op en subs e t f ( U ) of X suc h that f ( U ) ∩ Y = U . Set B = f [ A ]. Then |B | = λ b ecause f is bijectiv e. Supp ose C ∈ [ B ] κ and p is in the interior o f T C with resp ect to X . Then p is in the in terior of Y ∩ T C with resp ect to Y , in contradiction how we chose A . Thus, B is a lo cal h λ, κ i -splitter at p in X . By Lemma 5.3, χN t ( p, X ) ≤ κ .  Deﬁnition 5.15. F or all inﬁnite c a rdinals κ , let u ( κ ) denote the spa ce of uniform ultraﬁlters on κ . Theorem 5.16. F or e ach κ ≥ ω , ther e exists p ∈ u ( κ ) such that χN t ( p, u ( κ )) = ω and χ ( p, u ( κ )) = 2 κ . Pr o of. Let A b e a n indep endent family of subsets o f κ of s ize 2 κ . Set B = S F ∈ [ A ] ω { x ⊆ κ : ∀ y ∈ F | x \ y | < κ } . Since A is indepe ndent, w e may e xtend A to an ultr aﬁlter p on κ s uch that p ∩ B = ∅ . F or each x ⊆ κ , set x ∗ = { q ∈ u ( κ ) : x ∈ q } . Then { x ∗ : x ∈ A } is a lo cal h 2 κ , ω i -splitter at p . Since χ ( p, u ( κ )) ≤ 2 κ , it follows from Lemma 5 .3 that χN t ( p, u ( κ )) = ω and χ ( p, u ( κ )) = 2 κ .  Theorem 5.17. Su pp ose κ ≥ ω and X is a sp ac e s u ch that χ ( X ) = 2 κ and u ( κ ) emb e ds in X . Then ther e is an ω op -like lo c al b ase at some p oint in X . Henc e, χN t ( X ) = ω if X is homo gene ous. Pr o of. Let j embed u ( κ ) into X . By Theorem 5.16, there exists p ∈ u ( κ ) such that χN t ( p, u ( κ )) = ω and χ ( p, u ( κ )) = 2 κ . By Lemma 5.14, χN t ( j ( p ) , X ) = ω .  References [1] A. V. Arhangel ′ ski ˘ ı, T op olo g ic al homo geneity. T op olo gic al gr oups and their c ontinuous im- ages , Russian Math. 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