Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. I
aa r X i v : . [ m a t h . L O ] M a r COMBINATORIAL AND MODEL-THEORETICALPRINCIPLES RELATED TO REGULARITY OFULTRAFILTERS AND COMPACTNESS OFTOPOLOGICAL SPACES. I.
PAOLO LIPPARINI
Abstract.
We begin the study of the consequences of the exis-tence of certain infinite matrices. Our present application is tocompactness of products of topological spaces.
Our notation is fairly standard. See, e. g., [CN, KV, HNV] for un-explained notation.Ordinals are denoted by α, β, γ, . . .
Infinite cardinals are denoted by λ, µ, ν, κ, . . .
Inclusion is denoted by ⊆ , and ⊂ denotes strict inclusion.The minus operation between sets is denoted by \ , that is, X \ Y = { x ∈ X | x Y } .We assume the Axiom of Choice.If ( X α ) α<λ are topological spaces, then Q α<λ X α denotes their prod-uct with the Tychonoff topology, the smallest topology under whichthe canonical projections are continue maps.The λ -th power of a topological space X is the product Q α<λ X α ,where X α = X for all α ∈ λ .If κ, λ are infinite cardinals, a topological space is said to be [ κ, λ ] -compact if and only if every open cover by at most λ sets has a subcoverby less than κ sets.No separation axiom is needed to prove the results of the presentpaper.The following characterizations are old and well-known. See [L5,Section 3] for details, further references and further information about[ κ, λ ]-compactness. Proposition 1.
For every infinite regular cardinal κ and every topo-logical space X , the following are equivalent. Mathematics Subject Classification.
Primary 03E05, 54B10, 54D20 ; Sec-ondary 03E75.
Key words and phrases.
Infinite matrices, compactness of products of topologicalspaces.The author has received support from MPI and GNSAGA. We wish to expressedour gratitude to X. Caicedo for stimulating discussions and correspondence. (i) X is [ κ, κ ] -compact.(ii) Whenever ( U α ) α<κ is a sequence of open sets of X , such that U α ⊆ U α ′ for every α < α ′ , and such that S α<κ U α = X , then there isan α < κ such that U α = X .(iii) Whenever ( C α ) α<κ is a sequence of closed sets of X , such that C α ⊇ C α ′ for every α < α ′ , and such that T α<κ C α = ∅ , then there isan α < κ such that C α = ∅ .(iv) For every sequence ( x α ) α<κ of elements of X , there exists x ∈ X such that |{ α < κ | x α ∈ U }| = κ for every neighbourhood U of x .(v) (CAP κ ) Every subset Y ⊆ X with | Y | = κ has a complete accu-mulation point. Theorem 2.
Suppose that λ , µ are infinite regular cardinals, and κ isan infinite cardinal. Then the following conditions are equivalent.(a) There is a family ( B α,β ) α<µ,β<κ of subsets of λ such that:(i) For every β < κ , S α<µ B α,β = λ ;(ii) For every β < κ and α ≤ α ′ < µ , B α,β ⊆ B α ′ ,β ;(iii) For every function f : κ → µ there exists a finite subset F ⊆ κ such that | T β ∈ F B f ( β ) ,β | < λ .(b) Whenever ( X β ) β<κ is a family of topological spaces such that no X β is [ µ, µ ] -compact, then X = Q β<κ X β is not [ λ, λ ] -compact.(c) The topological space µ κ is not [ λ, λ ] -compact, where µ is endowedwith the topology whose open sets are the intervals [0 , α ) ( α ≤ µ ), and µ κ is endowed with the Tychonoff topology.Remark . In a sequel to this note we shall provide many more condi-tions equivalent to the conditions in Theorem 2. The same applies tothe conditions we shall introduce in Theorems 5 and 6.
Proof. (a) ⇒ (b). Let X, ( X β ) β<κ and ( B α,β ) α<µ,β<κ be as in the state-ment of the theorem.Since no X β is [ µ, µ ]-compact, and since µ is regular, by Condition(iv) in Proposition 1, for every β < κ there is a sequence { x α,β | α < µ } of elements of X β such that every x ∈ X β has a neighbourhood U β in X β such that |{ α < µ | x α,β ∈ U β }| < µ .We shall define a sequence ( y γ ) γ<λ of elements of X such that forevery z ∈ X there is a neighbourhood U in X of z such that |{ γ <λ | y γ ∈ U }| < λ , thus X is not [ λ, λ ]-compact, again by Condition (iv)in Proposition 1, and since λ is supposed to be regular.For γ < λ , let y γ = (( y γ ) β ) β<κ ∈ Q β<κ X β be defined by: ( y γ ) β = x α,β , where α is the first ordinal such that γ ∈ B α,β (such an ordinalexists by Condition (i)). OMBINATORIAL PRINCIPLES, COMPACTNESS OF SPACES 3
Suppose by contradiction that there is z ∈ X such that for everyneighbourhood U in X of z |{ γ < λ | y γ ∈ U }| = λ .Consider the components ( z β ) β<κ of z ∈ X = Q β<κ X β . Because ofthe way we have chosen the x α,β s, for each β < κ , z β has a neighbour-hood U β in X β such that |{ α < µ | x α,β ∈ U β }| < µ . For every β < κ ,fix some U β as above. For each β < κ , choose f ( β ) in such a way that µ > f ( β ) > sup { α < µ | x α,β ∈ U β } (this is possible since µ is regular,and |{ α < µ | x α,β ∈ U β }| < µ ).By Condition (iii) there is a finite F ⊆ κ such that | T β ∈ F B f ( β ) ,β | <λ . Let V = Q β<κ V β , where V β = X β if β F , and V β = U β if β ∈ F . V is a neighbourhood of z in X , since F is finite.For every γ < λ and β < κ , by definition, ( y γ ) β = x α,β , for some α such that γ ∈ B α,β . By the definition of f , if ( y γ ) β = x α,β ∈ U β then f ( β ) > α , thus γ ∈ B α,β ⊆ B f ( β ) ,β , by Condition (ii). We have provedthat, for every β < κ , { γ < λ | ( y γ ) β ∈ U β } ⊆ B f ( β ) ,β .Thus, by the definition of V , we have { 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 we have supposed that |{ γ < λ | y γ ∈ V }| = λ , for every neighbourhood V of z .(b) ⇒ (c) is trivial, since µ is not [ µ, µ ]-compact.(c) ⇒ (a). By Condition (iv) in Proposition 1 there exists a se-quence ( y γ ) γ<λ of elements in µ κ such that for every z ∈ µ κ there is aneighbourhood U in µ κ of z such that |{ γ < λ | y γ ∈ U }| < λ .For each γ < λ , y γ ∈ µ κ has the form y γ = (( y γ ) β ) β<κ . For 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), suppose that f : κ → µ . Let z ∈ µ κ be definedby z = ( f ( β )) β<κ . By the first paragraph, there is a neighbourhood U in µ κ of z such that |{ γ < λ | y γ ∈ U }| < λ .Arguing componentwise, this means that there are a finite set F ⊆ κ and, for each β ∈ F , neighbourhoods U β of f ( β ) in µ such that | T β ∈ F { γ < λ | ( y γ ) β ∈ U β }| < λ . Since any neighbourhood U β of f ( β )in µ contains [0 , f ( β ) + 1), we have that ( 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 .Thus, T β ∈ F B f ( β ) ,β = T β ∈ F { γ < λ | ( y γ ) β ≤ f ( β ) } has cardinality < λ . (cid:3) Remark . In the particular case λ = κ = µ + [L5, Lemma 14] statesthat Condition (a) in Theorem 2 is true, and, actually, we can get | F | =2 (the proof elaborates on a variation on a classical combinatorial deviceknown as an “Ulam matrix” [EU]). Proposition 15 in [L5] then goeson showing that, in the above particular case λ = κ = µ + , Condition COMBINATORIAL PRINCIPLES, COMPACTNESS OF SPACES (b) in Theorem 2 holds. Thus, modulo [L5, Lemma 14], Theorem 2generalizes [L5, Proposition 15]. Indeed, our proof of (a) ⇒ (b) inTheorem 2 is modelled after the proof of Proposition 15 in [L5].The main results proved in [L5] had been announced in [L4], wherefurther results similar to the ones presented here are stated. [C1, C2,L1, L2, L3] also contain related results. We plan to give a unifiedtreatment of all these results in a sequel to the present note.Theorem 2 can be generalized for box products.If ν is a cardinal, and ( X β ) β<κ is a family of topological spaces, thentheir product can be assigned the ✷ <ν topology, the topology a base ofwhich is given by all products ( Y β ) β<κ , where each Y β is an open subsetof X β , and |{ β < κ | Y β = X β }| < ν . The product of ( X β ) β<κ with the ✷ <ν topology shall be denoted by ✷ <νβ<κ X β . Theorem 5.
Suppose that λ , µ are infinite regular cardinals, and κ , ν are infinite cardinals.Then the following conditions are equivalent.(a) There is a family ( B α,β ) α<µ,β<κ of subsets of λ such that:(i) For every β < κ , S α<µ B α,β = λ ;(ii) For every β < κ and α ≤ α ′ < µ , B α,β ⊆ B α ′ ,β ;(iii) For every function f : κ → µ there exists a subset F ⊆ κ suchthat | F | < ν and | T β ∈ F B f ( β ) ,β | < λ .(b) Whenever ( X β ) β<κ is a family of topological spaces such that no X β is [ µ, µ ] -compact, then X = ✷ <νβ<κ X β is not [ λ, λ ] -compact.(c) The topological space µ κ is not [ λ, λ ] -compact, where µ is endowedwith the topology whose open sets are the intervals [0 , α ) ( α ≤ µ ), and µ κ is endowed with the ✷ <ν topology.Proof. The proof is identical to the proof of Theorem 2. (cid:3)
Notice that Theorem 5 generalizes Theorem 2, since the Tychonoffproduct is just the box product ✷ <ω . Hence Theorem 2 is the particularcase ν = ω of Theorem 5.We have an even more general version of the above theorems. Theorem 6.
Suppose that λ is an infinite regular cardinal, κ , ν areinfinite cardinals, and ( µ β ) β<κ are infinite regular cardinals.Then the following conditions are equivalent.(a) There is a family ( B α,β ) β<κ,α<µ β of subsets of λ such that:(i) For every β < κ , S α<µ β B α,β = λ ;(ii) For every β < κ and α ≤ α ′ < µ β , B α,β ⊆ B α ′ ,β ; OMBINATORIAL PRINCIPLES, COMPACTNESS OF SPACES 5 (iii) For every f ∈ Q β<κ µ β there 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 family of topological spaces such that forno β < κ X β is [ µ β , µ β ] -compact, then X = ✷ <νβ<κ X β is not [ λ, λ ] -compact.(c) The topological space ✷ <νβ<κ µ β is not [ λ, λ ] -compact, where, foreach β < κ , µ β is endowed with the topology whose open sets are theintervals [0 , α ) ( α ≤ µ β ).Proof. The proof is similar to the proof of Theorem 2. (cid:3)
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Dipartimento di Matematica, Viale della Ricerca Scientifica, IIUniversit`a de Roma (Tor Vergata), I-00133 ROME ITALY
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