Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. I

COMBINA TORIAL AND MODEL-THEORETICA L PRINCI PLES RELA TED TO REGULARI TY OF UL TRAFIL TERS AND COM P ACTNESS OF TOPOLOGICAL SP ACES. I. P AOLO LIPP ARINI Abstract. W e beg in the study of the cons equences of the exis- tence of certain infinite matrices. Our present application is to compactness of pro ducts of top olog ical spaces. Our notation is fairly standard. See, e. g., [CN, KV, HNV] fo r un- explained notation. Ordinals are de noted by α, β , γ , . . . Infin ite cardinals are denoted by λ, µ, ν , κ, . . . Inc lusion is denoted b y ⊆ , and ⊂ denotes strict inclusion. The min us op eration b etw een sets is denoted b y \ , that is, X \ Y = { x ∈ X | x 6∈ Y } . W e a ssume the Axiom of Choice. If ( X α ) α<λ are top olo g ical spaces, then Q α<λ X α denotes their pro d- uct with the T yc honoff to p ology , t he smallest top ology under whic h the canonical pro jections are con t inue maps. The λ -th p ower of a top ological space X is the pro duct Q α<λ X α , where X α = X for all α ∈ λ . If κ, λ are infinite cardinals, a top ological space is said t o b e [ κ, λ ] - c omp act if and only if ev ery op en co v er b y at most λ sets has a sub cov er b y less tha n κ sets. No separation axiom is needed t o prov e the results of the presen t pap er. The follo wing ch aracterizations are old and w ell-known. See [L5 , Section 3] for details, further references and further informat ion about [ κ, λ ]-compactness. Prop osition 1. F or every infinite r e gular c ar dinal κ and every top o- lo gic al sp a c e X , the fol lowing ar e e quivalent. 2000 Mathematics Subje ct Classific ation. Primar y 03E05, 54B10, 54D20 ; Sec- ondary 03 E75. Key wor ds and phr ases. Infinite matrices, compactness of products of top ologica l spaces. The author has received support from MPI and GNSAGA . W e wish to expressed our gra titude to X. C a icedo for stimulating discussions and co rresp ondence. 1 2 COMBINA TORIAL PRINCIPLES, COMP ACTNESS OF S P ACES (i) X is [ κ, κ ] -c omp act. (ii) Whenever ( U α ) α<κ is a se quenc e of op en sets o f X , such that U α ⊆ U α ′ for every α < α ′ , and such that S α<κ U α = X , then ther e is an α < κ such that U α = X . (iii) Whenever ( C α ) α<κ is a se quenc e of close d sets of X , such that C α ⊇ C α ′ for eve ry α < α ′ , and s uch that T α<κ C α = ∅ , then ther e is an α < κ such that C α = ∅ . (iv) F or every se quenc e ( x α ) α<κ of el e m ents of X , t her e ex i s ts x ∈ X such that | { α < κ | x α ∈ U }| = κ for every neighb ourho o d U of x . (v) (CAP κ ) Every subset Y ⊆ X with | Y | = κ h as a c omplete ac cu- mulation p oin t. Theorem 2. Supp ose that λ , µ ar e infinite r e gular c ar dinals, and κ is an in fi nite c ar dinal. Then the fol lowing c onditions a r e e quivalent. (a) Ther e is a family ( B α,β ) α<µ,β <κ of subsets of λ such that: (i) F or every β < κ , S α<µ B α,β = λ ; (ii) F or eve ry β < κ and α ≤ α ′ < µ , B α,β ⊆ B α ′ ,β ; (iii) F or every function f : κ → µ ther e exists a finite subset F ⊆ κ such that | T β ∈ F B f ( β ) ,β | < λ . (b) Whenever ( X β ) β <κ is a famil y of top olo gic al sp ac es such that no X β is [ µ, µ ] -c omp act, then X = Q β <κ X β is not [ λ, λ ] -c omp act. (c) The top olo gic al sp ac e µ κ is not [ λ, λ ] -c om p act, wher e µ is endowe d with the top olo g y whose op en sets a r e the intervals [0 , α ) ( α ≤ µ ), and µ κ is endowe d with the T ychonoff top olo gy. R emark 3 . In a seque l to this note w e shall pro vide many more condi- tions equiv alen t to the conditions in Theorem 2. The same applies to the conditions w e shall in tro duce in Theorems 5 and 6. Pr o of. (a) ⇒ (b). Let X , ( X β ) β <κ and ( B α,β ) α<µ,β <κ b e as in the state- men t of the theorem. Since no X β is [ µ, µ ]- compact, and since µ is regular, b y Condition (iv) in Prop osition 1, for ev ery β < κ there is a sequence { x α,β | α < µ } of elemen ts of X β suc h that ev ery x ∈ X β has a neighbourho o d U β in X β suc h that |{ α < µ | x α,β ∈ U β }| < µ . W e shall define a sequence ( y γ ) γ <λ of elemen ts o f X suc h that for ev ery z ∈ X t here is a neigh b ourho o d U in X of z suc h that |{ γ < λ | y γ ∈ U } | < λ , th us X is not [ λ, λ ]-compact, a gain b y Condition (iv) in Prop osition 1, and since λ is supp osed to be regular. F or γ < λ , let y γ = (( y γ ) β ) β <κ ∈ Q β <κ X β b e defined b y: ( y γ ) β = x α,β , where α is the first ordinal suc h that γ ∈ B α,β (suc h an ordinal exists b y Condition (i)). COMBINA TORIAL PRINCIPLES, COMP ACTNESS OF S P ACES 3 Supp ose by con tradiction that there is z ∈ X suc h that fo r ev ery neigh b o urho o d U in X of z | { γ < λ | y γ ∈ U }| = λ . Consider the comp o nents ( z β ) β <κ of z ∈ X = Q β <κ X β . Because of the w a y w e ha ve c hosen the x α,β s, for eac h β < κ , z β has a neigh b o ur- ho o d U β in X β suc h that |{ α < µ | x α,β ∈ U β }| < µ . F or ev ery β < κ , fix some U β as ab ov e. F or eac h β < κ , c ho ose f ( β ) in s uc h a w ay that µ > f ( β ) > sup { α < µ | x α,β ∈ U β } (this is p ossible since µ is regular, and |{ α < µ | x α,β ∈ U β }| < µ ). By C ondition (iii) there is a finite F ⊆ κ suc h that | T β ∈ F B f ( β ) ,β | < λ . Let V = Q β <κ V β , where V β = X β if β 6∈ F , and V β = U β if β ∈ F . V is a neigh b ourho o d of z in X , since F is finite. F or ev ery γ < λ and β < κ , b y definition, ( y γ ) β = x α,β , for some α suc h that γ ∈ B α,β . By the definition of f , if ( y γ ) β = x α,β ∈ U β then f ( β ) > α , thu s γ ∈ B α,β ⊆ B f ( β ) ,β , by Condition (ii). W e ha ve pro v ed that, for ev ery β < κ , { γ < λ | ( y γ ) β ∈ U β } ⊆ B f ( β ) ,β . Th us, by the definition of V , w e ha v e { γ < λ | y γ ∈ V } = T β ∈ F { γ < λ | ( y γ ) β ∈ U β } ⊆ T β ∈ F B f ( β ) ,β . Hence |{ γ < λ | y γ ∈ V }| ≤ | T β ∈ F B f ( β ) ,β | < λ . This is a contradiction, since w e ha ve supp osed that | { γ < λ | y γ ∈ V }| = λ , for ev ery neigh b ourho o d V of z . (b) ⇒ (c) is trivial, since µ is not [ µ, µ ]-compact. (c) ⇒ (a ). By Condition ( iv) in Prop osition 1 there exists a se- quence ( y γ ) γ <λ of elemen ts in µ κ suc h that for ev ery z ∈ µ κ there is a neigh b o urho o d U in µ κ of z suc h tha t |{ γ < λ | y γ ∈ U }| < λ . F or each γ < λ , y γ ∈ µ κ has the form y γ = (( y γ ) β ) β <κ . F or α < µ and β < κ define B α,β = { γ < λ | ( y γ ) β ≤ α } . Conditions (i) and (ii) in (a) trivially hold. As for Condition (iii), supp ose that f : κ → µ . Let z ∈ µ κ b e defined b y z = ( f ( β )) β <κ . By the first paragra ph, there is a neigh b ourho od U in µ κ of z suc h tha t |{ γ < λ | y γ ∈ U }| < λ . Arguing comp onen t wise, this means that there are a finite set F ⊆ κ and, for eac h β ∈ F , neighbourho ods U β of f ( β ) in µ suc h that | T β ∈ F { γ < λ | ( y γ ) β ∈ U β }| < λ . Since an y neighbourho o d U β of f ( β ) in µ con ta ins [0 , f ( β ) + 1) , w e ha ve tha t ( y γ ) β ≤ f ( β ) implies that ( y γ ) β ∈ U β . Hence also | T β ∈ F { γ < λ | ( y γ ) β ≤ f ( β ) }| < λ . Th us, T β ∈ F B f ( β ) ,β = T β ∈ F { γ < λ | ( y γ ) β ≤ f ( β ) } has car dina lity < λ .  R emark 4 . In the particular case λ = κ = µ + [L5, Lemma 14] states that Condition (a) in Theorem 2 is true, and, actually , w e can get | F | = 2 (the pro of elab orates on a v ariation on a classical com binatoria l devic e kno wn a s an “Ulam matrix” [EU]). Prop osition 15 in [L5] then go es on sho wing that, in t he a b o v e particular case λ = κ = µ + , Condition 4 COMBINA TORIAL PRINCIPLES, COMP ACTNESS OF S P ACES (b) in Theorem 2 holds. Thus, mo dulo [L 5 , Lemma 1 4], Theorem 2 generalizes [L5, Prop o sition 15]. Inde ed, our pro of of (a) ⇒ (b) in Theorem 2 is mo delled after the pro of o f Prop osition 15 in [L5]. The main results prov ed in [L5] had b een a nnounced in [L4], where further results similar to the ones presen t ed here are stated. [C1, C2, L1, L2, L3] also contain related results. W e plan to give a unified treatmen t o f all these results in a sequel to the presen t note. Theorem 2 can b e generalized fo r b o x pro ducts. If ν is a cardinal, and ( X β ) β <κ is a family of topolo gical spaces, then their pro duct can b e assigned the ✷ <ν top ology , the to p ology a base of whic h is giv en by all pro ducts ( Y β ) β <κ , where each Y β is an op en subset of X β , and |{ β < κ | Y β 6 = X β }| < ν . The pro duct o f ( X β ) β <κ with t he ✷ <ν top ology shall b e denoted by ✷ <ν β <κ X β . Theorem 5. Supp ose that λ , µ ar e infinite r e gular c ar dinals, and κ , ν ar e infin ite c ar dinals. Then the fol lowing c onditions a r e e quivalent. (a) Ther e is a family ( B α,β ) α<µ,β <κ of subsets of λ such that: (i) F or every β < κ , S α<µ B α,β = λ ; (ii) F or eve ry β < κ and α ≤ α ′ < µ , B α,β ⊆ B α ′ ,β ; (iii) F or every function f : κ → µ ther e exists a subset F ⊆ κ such that | F | < ν and | T β ∈ F B f ( β ) ,β | < λ . (b) Whenever ( X β ) β <κ is a famil y of top olo gic al sp ac es such that no X β is [ µ, µ ] -c omp act, then X = ✷ <ν β <κ X β is not [ λ, λ ] - c omp act. (c) The top olo gic al sp ac e µ κ is not [ λ, λ ] -c om p act, wher e µ is endowe d with the top olo g y whose op en sets a r e the intervals [0 , α ) ( α ≤ µ ), and µ κ is endowe d with the ✷ <ν top olo gy. Pr o of. The pro of is iden tical to the pro of of Theorem 2.  Notice t hat Theorem 5 generalizes Theorem 2, since the T ychonoff pro duct is j ust the b ox pro duct ✷ <ω . Hence Theorem 2 is the particular case ν = ω of Theorem 5. W e hav e an ev en more general vers ion of the ab o v e theorems. Theorem 6 . Supp ose that λ is an infinite r e gular c ar dinal, κ , ν ar e infinite c a r din als, and ( µ β ) β <κ ar e infinite r e gular c ar dinals. Then the fol lowing c onditions a r e e quivalent. (a) Ther e is a family ( B α,β ) β <κ,α<µ β of subsets of λ such that: (i) F or every β < κ , S α<µ β B α,β = λ ; (ii) F or eve ry β < κ and α ≤ α ′ < µ β , B α,β ⊆ B α ′ ,β ; COMBINA TORIAL PRINCIPLES, COMP ACTNESS OF S P ACES 5 (iii) F or every f ∈ Q β <κ µ β ther e exi s ts a subse t F ⊆ κ such that | F | < ν and | T β ∈ F B f ( β ) ,β | < λ . (b) Whenever ( X β ) β <κ is a family of top olo gic al sp ac es such that for no β < κ X β is [ µ β , µ β ] -c omp act, then X = ✷ <ν β <κ X β is not [ λ, λ ] - c omp act. (c) The top olo gic al sp ac e ✷ <ν β <κ µ β is no t [ λ, λ ] -c omp act, wher e, for e ach β < κ , µ β is endowe d with the top olo gy whose op en sets ar e the intervals [0 , α ) ( α ≤ µ β ). Pr o of. 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