aa r X i v : . [ m a t h . GN ] A ug COMPACT SPACES WITH A P -BASE ALAN DOW AND ZIQIN FENG
Abstract.
In the paper, we investigate (scattered) compact spaces with a P -base for some poset P . More specifically, we prove that any compact spacewith an ω ω -base is metrizable and any scattered compact space with an ω ω -base is countable under the assumption ω < b . These give positive solutionsto Problems 8.6.9 and 8.7.7 in [2]. Using forcing, we also prove that in a modelof ω < b , there is a non-first-countable compact space with a P -base for someposet P with calibre ω . Introduction
Let P be a partially ordered set. A topological space X is defined to have aneighborhood P -base at x ∈ X if there exists a neighborhood base ( U p [ x ]) p ∈ P at x such that U p [ x ] ⊂ U p ′ [ x ] for all p ≥ p ′ in P . We say that a topological space has a P -base if it has a neighborhood P -base at each x ∈ X .We will use Tukey order to compare the cofinal complexity of posets. The Tukeyorder [19] was originally introduced, early in the 20th century, as a tool to under-stand convergence in general topological spaces, however it was quickly seen to havebroad applicability in comparing partial orders. Given two directed sets P and Q ,we say Q is a Tukey quotient of P , denoted by P ≥ T Q , if there is a map φ : P → Q carrying cofinal subsets of P to cofinal subsets of Q . In our context, where P and Q are both Dedekind complete (every bounded subset has a least upper bound),we have P ≥ T Q if and only if there is a map φ : P → Q which is order-preservingand such that φ ( P ) is cofinal in Q . If P and Q are mutually Tukey quotients, wesay that P and Q are Tukey equivalent, denoted by P = T Q . It is straightforwardto see that a topological space X has a P -base if and only if T x ( X ) ≤ T P for each x ∈ X , here, T x ( X ) = { U : U is an open neighborhood of x } .Topological spaces and function spaces with an ω ω -base were systematically stud-ied in [2]. Lots of work also have been done with the topological algebra with an ω ω -base in [3], [8], [15], and [18]. In this paper we investigate the Tukey reductionof a P -base in some (scattered) compact spaces with P satisfying some Calibreconditions. This paper is organized in the following way.In Section 3, we show that if P has Calibre ω , then any compact space witha P -base is countable tight. Furthermore, we prove that if a compact space withcountable tightness has a K ( M )-base for some separable metric space M , thenit is first-countable. As a corollary, any compact space with an ω ω -base is first-countable under the assumption ω < b . This gives a positive answer to Problem8.7.7 in [2]. In Section 4, we address Problem 8.6.9 in [2] positively by showing that Date : August 12, 2020.2010
Mathematics Subject Classification.
Key words and phrases. (Scattered) compact spaces, scattered height, Tukey order, Calibre ω ,Calibre ( ω , ω ), convergent free sequence, ω ω -base, countable tightness . any scattered compact space with an ω ω -base is countable under the assumption ω < b . It is natural to ask whether under the assumption ω < b any compactwith a P -base is first-countable if P satisfies some Calibre properties, for example,Calibre ω . In Section 5, we prove that in a model of Martin’s Axiom in which ω < b , there is a non-first-countable compact space with a P -base for some poset P with calibre ω . 2. Preliminaries
For any separable metric space M , K ( M ) is the collection of compact subsets of M ordered by set-inclusion. Fremlin observed that if a separable metric space M islocally compact, then K ( M ) = T ω . Its unique successor under Tukey order is theclass of Polish but not locally compact spaces. For M in this class, K ( M ) = T ω ω where ω ω is ordered by f ≤ g if f ( n ) ≤ g ( n ) for each n ∈ ω . In [9], Gartside andMamataleshvili constructed a 2 c -sized antichain in K ( M ) = {K ( M ) : M ∈ M} where M is the set of separable metric spaces.Let P be a directed poset, i.e. for any points p, p ′ ∈ P , there exists a point q ∈ P such that p ≤ q and p ′ ≤ q . A subset C of P is cofinal in P if for any p ∈ P , thereexists a q ∈ C such that p ≤ q . Then cof( P ) = min {| C | : C is cofinal in P } . Wealso define add( P ) = min {| Q | : Q is unbounded in P } . For any f, g ∈ ω ω , we saythat f ≤ ∗ g if the set { n ∈ ω : f ( n ) > g ( n ) } is finite. Then b = add( ω ω , ≤ ∗ ) and d = cof( ω ω , ≤ ∗ ). See [1] for more information about small cardinals.Let κ ≥ µ ≥ λ be cardinals. We say that a poset P has calibre ( κ, µ, λ ) if forevery κ -sized subset S of P there is a µ -sized subset S such that every λ -sizedsubset of S has an upper bound in P . We write calibre ( κ, µ, µ ) as calibre ( κ, µ )and calibre ( κ, κ, κ ) as calibre κ . It is known that K ( M ) has Calibre ( ω , ω ) for anyseparable metric space M , hence so does ω ω . Under the assumption ω < b , ω isa Tukey quotient of ω ω . Furthermore, under the assumption ω < b , the poset ω ω has Calibre ( ω , ω , ω ), i.e. Calibre ω . We will use this fact in several places ofthis paper.It is clear that if P ≤ T Q and Q ≤ T R then P ≤ T R for any posets P, Q, and R . So we get the following proposition. Proposition 2.1.
Let P and Q be posets such that P ≤ T Q . Then if a space X has a neighborhood P -base at x ∈ X , then X also has a neighborhood Q -base at x .Hence, any space with a P -base also has a Q -base. Proposition 2.2. If X has a P -base, then any subspace of X also has a P -base. Proposition 2.3.
Let P be a poset with ω ≤ T P and P = T ω × P . Then thespace ω + 1 has a P -base.Proof. For each α < ω , the space ω + 1 has a countable local base at α . Hence T α ( ω + 1) ≤ T P due to the fact that P = T ω × P .Let φ be a map from P to ω which carries confinal subsets of P to confinalsubsets of ω . Then we define a map ψ from P to T ω ( ω +1) by ψ ( p ) = ( φ ( p ) , ω ] foreach p ∈ P . Clearly ψ carries confinal subsets of P to confinal subset of T ω ( ω + 1).Hence the space ω +1 has a neighborhood P -base at ω . This finishes the proof. (cid:3) As a result of b ≤ T ω ω , the space ω + 1 has an ω ω -base under the assumption ω = b . Gartside and Mamatelashvili in [10] proved that ω ω × ω ≤ T K ( Q ) ≤ T ω ω × [ ω ] <ω , here Q is the space of rationals. Hence, we have the following result. OMPACT SPACES WITH A P -BASE 3 Corollary 2.4.
The space ω + 1 has a K ( Q ) -base. A generalization of G δ -diagonals is P -diagonals for some poset P . A collection C of subsets of a space X is P -directed if C can be represented as { C p : p ∈ P } such that C p ⊆ C p ′ whenever p ≤ p ′ . We say X has a P -diagonal if X \ ∆ has a P -directed compact cover, where ∆ = { ( x, x ) : x ∈ X } . The second author showedthat any compact space with a K ( Q )-diagonal is metrizable in [7] and S´anchezproved that the same result holds for any compact space with a K ( M )-diagonal forsome separable metric space M in [17]. Here, we include two results about spaceswith (or without) P -diagonal giving that P satisfies some Calibre properties. Proposition 2.5.
Let P be a poset with Calibre ( ω , ω ) . The space ω + 1 doesn’thave a P -diagonal.Proof. Suppose that ω + 1 has a P -diagonal, i.e., a P -ordered compact covering { K p : p ∈ P } of ( ω + 1) \ ∆ Choose α γ and β γ in ω for γ ∈ ω such that α < β < α < β < . . . < α γ < β γ < . . . . Let p γ in P be such that ( α γ , β γ ) ∈ K p γ for each γ ∈ ω . By Calibre ( ω , ω ),there are p in P and γ n ∈ ω , n ∈ ω such that γ < γ < . . . and K p γn ⊂ K p foreach n ∈ ω . Then ( δ, δ ) ∈ K p , where δ = sup { α γ n : n ∈ ω } . This contradictionfinishes the proof. (cid:3) Proposition 2.6.
Let P be a poset with Calibre ω . Any compact space with a P -diagonal has countable tightness.Proof. Let { K p : p ∈ P } be a P -ordered compact covering of X \ ∆. Supposethat X has uncountable tightness. Then, X has a free sequence of length ω , hencea convergence free sequence of length ω by [12]. Let { x α : α < ω } be such asequence and x ∗ the limit point.Choose α γ and β γ in ω for γ ∈ ω such that α < β < α < β < . . . <α γ < β γ < . . . . For each γ ∈ ω , fix p γ ∈ P such that ( x α γ , x β γ ) ∈ K p γ . Since P has Calibre ω , there is an uncountable subset γ τ of ω with τ < ω such that p γ τ is bounded above by p ∗ ∈ P . Hence, K p γτ ⊆ K p ∗ for each τ < ω , furthermore,( x α γτ , x β γτ ) ∈ K p ∗ . Since { x α : α < ω } is a convergent sequence with limit point x ∗ , the subsequences x α γτ and x β γτ both converge to x ∗ , hence, ( x ∗ , x ∗ ) ∈ K p ∗ which contradicts with the fact that K p ∗ ⊂ X \ ∆. This contradiction finishes theproof. (cid:3) Compact Spaces
In this section, we study compact spaces possessing a P -base with P satisfyingsome Calibre property. Mainly, we investigate the problem whether each compactHausdorff space with an ω ω -base first countable under the assumption ω < b . Westart with some ZFC result about tightness of the spaces with a P -base. Theorem 3.1.
Let κ be an uncountable regular cardinal and P be a poset withCalibre κ . If X is a compact Hausdorff space with a P -base, then t ( X ) < κ .Proof. Assume, for contradiction, that t ( X ) ≥ κ . Then, X has a free sequence oflength κ . Hence, by [12], X has a convergent sequence of length κ . Let { x α : α < κ } be such a sequence and x ∗ be the limit point. Let S = { x α : α < κ } ∪ { x ∗ } . Noticethat for any unbounded subset { α γ : γ < κ } , x ∗ is the limit point of { x α γ : γ < κ } . A. DOW AND Z. FENG
Fix a neighborhood base { B p : p ∈ P } at x ∗ . It is straightforward to see that S \ B p has size < κ for each p ∈ P .For each α < κ , choose p α ∈ P such that x α / ∈ B p α . Let P ′ = { p α : α ∈ κ } .If the cardinality of P ′ is < κ , there exists a p α ∈ P ′ such that S \ B p α has size κ which is a contradiction. Hence, P ′ have cardinality κ . Since P has Calibre κ ,there is a κ -sized subset P ′′ of P ′ which is bounded above. List P ′′ as { p α γ : γ < κ } and pick p ∗ to be the upper bound of P ′′ . Then, S \ B p ∗ = { x α γ : γ < κ } which isa contradiction. This finishes the proof. (cid:3) Corollary 3.2.
Let P be a poset with Calibre ω . Each compact Hausdorff spacewith a P -base is countable tight. A poset P has Calibre ( ω , ω ) if it has Calibre ω . It is showed in [16] thatfor a separable metric space M the poset K ( M ) has Calibre ω if it has Cali-bre ( ω , ω , ω ). Hence it is natural to ask whether a compact space has countabletightness if it has a P -base with P having Calibre ( ω , ω ). The following exampleshows that the answer is negative. So the result above is ‘optimal’ in terms ofthe Calibre complexity of posets having the form K ( M ) with M being a separablemetric space. Example 3.3.
There is a poset P with Calibre ( ω , ω ) and a compact space X witha P -base, but t ( X ) > ω .Proof. Let P be K ( Q ) which clearly has Calibre ( ω , ω ). From proposition 2.4, thespace ω + 1 has a P -base, but its tightness is uncountable. (cid:3) Again, since ω ω has Calibre ω under the assumption ω < b , we obtain thefollowing result about spaces with an ω ω -base. Corollary 3.4.
Assume that ω < b . Each compact Hausdorff space with an ω ω -base is countable tight. It is folklore that any GO-space with countable tightness is first countable.Hence, applying Corollary 3.4 to compact GO-spaces, we obtain the following re-sults.
Corollary 3.5.
Let P be a poset with Calibre ω . Each compact GO-space has a P -base if and only if it is first countable. The following example shows that the result above doesn’t hold for general GO-spaces.
Example 3.6.
There is a poset P with Calibre ω such that there exists a GO-spacewith a P -base and uncountable tightness.Proof. Consider the set ω + 1 in the ordinal order. Let T be the topology on ω + 1 such that every point except ω is isolated and a base at ω is { ( α, ω ] : α <ω } . So the space ( ω + 1 , T ) is a non-first-countable GO-space and clearly has aneighborhood ω -base at ω . It is straightforward to verify that the poset ω hasCalibre ω since every ω -sized subset is bounded above. (cid:3) The result below was proved in [2] through a different approach. We obtain ithere as a result of ω ω having Calibre ω under the assumption ω < b . Corollary 3.7.
Assume that ω < b . Each compact GO-space has an ω ω -base ifand only if it is first countable. OMPACT SPACES WITH A P -BASE 5 It is natural to ask as in [2] (Problem 8.7.7) whether the same result holds forany compact space. Next, we’ll give a positive answer to this problem by showingthat any compact space with a P -base is first countable if P = K ( M ) for someseparable metric space M has Calibre ω .First, we show that any compact space with countable tightness is first countableif it has a P -base and P = K ( M ) for some separable metric space. We use the ideasand techniques from [5]. Theorem 3.8.
Let P = K ( M ) for some separable metric space M . If X is acompact space with countable tightness and has a P -base, then X is first-countable.Proof. Fix x ∈ X and an open P -base { U p [ x ] : p ∈ P } at x . For each p ∈ P , let K p = X \ U p [ x ]. Then, { K p : p ∈ P } is a P -directed compact cover of X \ { x } .For any separable metric space M , P = K ( M ) is also separable metrizable, hencesecond countable. Also if p n converges to p in P , then p ∗ = p ∪ ( S { p n } ) is alsocompact, hence it is an element in P with p n ⊂ p ∗ and p ⊂ p ∗ .Fix a countable base { B n : n ∈ ω } of P . For each n ∈ ω , define L n = S { K p : p ∈ B n } . And for each p ∈ P , define C ( p ) = T { L n : p ∈ B n } . Fix p ∈ P and adeceasing local base { B pn i : i ∈ ω } ⊆ { B n : n ∈ ω } at p . Define L pn i = S { K q : q ∈ B pn i } and C ′ ( p ) = T { L pn i : i ∈ ω } . It is straightforward to verify that C ′ ( p ) = C ( p ).First, we claim that x is not in the closure of C ( p ) for all p ∈ P . Fix p ∈ P . Bythe countable tightness of X , it suffices to show that if { y i : i ∈ ω } is a subset of C ( p ), then x is not in the closure of { y i : n ∈ ω } . Let { y i : n ∈ ω } be a subset of C ( p ). For each i ∈ ω , choose q i ∈ B pn i so that y i ∈ K q i . Clearly { q i : i ∈ ω } is asequence converging to p , hence p ∗ = p ∪ ( S { q i : i ∈ ω } ) is an element in P with y i ∈ K p ∗ which implies that x is not in the closure of { y i : n ∈ ω } .Then, we claim that for each p ∈ P , there is an i ∈ ω such that x is not inthe closure of L pn i . Fix p ∈ P . Choose any open set U such that C ( p ) ⊂ U and x / ∈ U . It suffices to prove that there is an i so that L pn i ⊂ U . Suppose not. Choose y i ∈ L pn i \ U for each i ∈ ω . Then for each i ∈ ω , choose q i ∈ B pn i so that y i ∈ K q i .Define p ∗ i = p ∪ ( S { q j : j ≥ i } ). Hence { y j : j ≥ i } ⊂ K p ∗ i . Clearly, p ∗ i ∈ B pn i ,hence K p ∗ i ⊂ L pn i for each i ∈ ω . Hence T { K p ∗ i : i ∈ ω } ⊆ C ( p ). Then, all the limitpoints of { y i : i ∈ ω } are in C ( p ) which contradicts with C ( h ) ⊂ U and the choicesof { y i : i ∈ ω } .Finally, we prove that the family L = { L n : x / ∈ L n } is a cover of X \ { x } ,furthermore, { B : B = X \ L for some L ∈ L} is a local base at x . For each p ∈ P ,there is an i ∈ ω such that that x is not in the closure of L pn i . Hence L pn i ∈ L .Since K p ⊂ L pn i , this completes the proof. (cid:3) Theorem 3.9.
Let P = K ( M ) for some separable metric space M such that P hasCalibre ω . Any compact space X with a P -base is first-countable.Proof. By Corollary 3.4, X has countable tightness since P has Calibre ω . Thenby Theorem 3.8, X is first-countable. (cid:3) Then using the fact that ω ω has Calibre ω under the assumption ω < b , weget a positive answer to Problem 8.7.7 in [2]. Corollary 3.10.
Assume ω < b . A compact space has an ω ω -base if and only ifit is first countable. A. DOW AND Z. FENG Scattered Compact Spaces
We recall that a topological space X is scattered if each non-empty subspace of X has an isolated point. The complexity of a scattered space can be determinedby the scattered height.For any subspace A of a space X , let A ′ be the set of all non-isolated points of A .It is straightforward to see that A ′ is a closed subset of A . Let X (0) = X and define X ( α ) = T β<α ( X ( β ) ) ′ for each α >
0. Then a space X is scattered if X ( α ) = ∅ for some ordinal α . If X is scattered, there exists a unique ordinal h ( x ) such that x ∈ X ( h ( x )) \ X ( h ( x )+1) for each x ∈ X . The ordinal h ( X ) = sup { h ( x ) : x ∈ X } is called the scattered height of X and is denoted by h ( X ). It is known that anycompact scattered space is zero-dimensional. Also, it is straightforward to showthat for any compact scattered space X , X ( h ( x )) is a non-empty finite subset. Theorem 4.1.
Let P be a poset with Calibre ω and X a scattered compact spacewith a P -base. Then X is countable.Proof. If h ( X ) = 0, then X is countable because it is compact.Assume h ( X ) = α and any compact scattered space with scattered height < α is countable. Since X is compact, X ( α ) is a nonempty finite subset of X . List X ( α ) = { x , . . . , x n } . For each i ∈ { , . . . , n } , take a closed and open neighborhood U i of x i with U i ∩ X ( α ) = { x i } . Then X \ S { U i : i = 1 , . . . , n } is a scatteredcompact space with scattered height < α , hence it is countable by the assumption.So it is sufficient to show that U i is countable for each i = 1 , . . . , n .Fix i ∈ { , . . . , n } . Consider the subspace Y = U i ∩ X . By proposition 2.2, Y has a neighborhood P -base { B p : p ∈ P } at { x i } . For each p ∈ P , Y \ B p is acompact subspace with scattered height < α , hence is countable by the inductiveassumption.Assume that Y is uncountable. Take an uncountable subset { y α : α < ω } of Y \ { x i } . For each α < ω , we choose p α ∈ P such that y α / ∈ B p α .If { p α : α < ω } is countable, there is a p ∗ ∈ { p α : α } such that there isan uncountable subset D of { y α : α < ω } such that D ⊂ Y \ B p ∗ which is acontradiction.If { p α : α < ω } is uncountable, then it has an uncountable subset P ′ which isbounded above using the Calibre ω property of P . List P ′ = { p α γ : γ < ω } . Let p ∗ be an upper bound of P ′ . Then we have that y α γ / ∈ B p ∗ for each γ < ω . Thisalso contradicts with the fact that Y \ B p ∗ is countable. This finishes the proof. (cid:3) Using the same approach we obtain the following example.
Example 4.2.
The one point Lindel¨ofication of uncountably many points doesn’thave a P -base if P has Calibre ω , hence, under the assumption ω < b , it doesn’thave an ω ω -base. The example above uses the fact that under the assumption ω < b , the poset ω ω has Calibre ω . Furthermore, using Theorem 4.1, we obtain the following resultwhich answers Problem 8.6.9 in [2] positively. This also gives a positive answer toProblem 8.6.8 in the same paper. Corollary 4.3.
Assume ω < b . Any scattered compact space with an ω ω -base iscountable, hence metrizable. OMPACT SPACES WITH A P -BASE 7 It was proved in [2] that any compact spaced with an ω ω -base and finite scatteredheight is countable, hence metrizable. Next, we show that the same result holds forany compact space with a P -base and finite scattered height if P has Calibre ( ω , ω ). Theorem 4.4.
Let P be a poset with Calibre ( ω , ω ) and X a compact Hausdorffspace with a P -base and finite scattered height. Then X is countable, hence metriz-able.Proof. Fix a natural number n >
0. Assume that any compact Hausdorff spacewith scattered height < n is countable. Let X be a compact Hausdorff space withscattered height n . We’ll show that X is countable. As discussed in the proofof Theorem 4.1, we could assume that X ( n ) is a singleton, denoted by x , withoutloss of generality. Suppose, for contradiction, that X is uncountable. Define m to be the greatest natural number such that X ( m ) is uncountable and X ( m +1) iscountable. Then there are two cases: 1. m = n −
1; 2. m < n −
1. We will obtaincontradictions in both cases.First assume that m = n −
1. Then we fix a neighborhood P -base { B p : p ∈ P } at x . For each p ∈ P , X ( m ) \ B p is finite as X is compact and B p is open. Pick anuncountable subset { x α : α < ω } of X ( m ) with x α = x for all α < ω . For each α < ω , there is a p α ∈ P such that x α / ∈ B p α . If { p α : α < ω } is countable, thenthere exists p ∗ ∈ { p α : α < ω } such that X ( m ) \ B p ∗ is uncountable which is acontradiction. If { p α : α < ω } is uncountable, we can find a countable boundedsubset { p α n : n ∈ ω } of { p α : α < ω } using the Calibre ( ω , ω ) property of P . Letthe upper bound of { p α n : n ∈ ω } be p ∗ . Then, x α n / ∈ B p ∗ for each n ∈ ω . This isa contradiction.Now we assume that m < n −
1. Then X ( m +1) \ { x } is countable which can belisted as { x ℓ : ℓ ∈ ω } . For each ℓ , pick a closed and open neighborhood U ℓ of x ℓ .Then for each ℓ < ω , U ℓ is a compact subspace with scattered height < n , hence iscountable. Therefore, X ( m ) \ S { U ℓ : ℓ ∈ ω } is uncountable. Pick an uncountablesubset S = { x α : α < ω } of X ( m ) \ ( { x } ∪ ( S { U ℓ : ℓ ∈ ω } )). Fix a neighborhood P base { B p : p ∈ P } at x . For each p ∈ P , S \ B p is finite. Similarly as in theproof of case 1, we could obtain a contradiction using the Calibre ( ω , ω ) propertyof P . (cid:3) The result above doesn’t hold for compact space with uncountable scatteredheight since the space ω + 1 has a K ( Q )-base. However, we don’t know the answerto the following problem. Question 4.5.
Assume that ω = b . Let P be a poset with Calibre ( ω , ω ) and X be any compact Hausdorff space with a P -base and countable scattered height. Is X countable? Calibre ω and non-first-countable compact space We prove that there is a model of Martin’s Axiom in which there is a compactspace that has a P -base for a poset P with Calibre ω . This space will be the spaceconstructed by Juhasz, Koszmider, and Soukup in the paper [11]. This article [11]shows there is a forcing notion that forces the existence of a first-countable initially ω -compact locally compact space of cardinality ω whose one-point compactifica-tion has countable tightness. We must prove that there is a poset P as above. Wemust also show that extra properties of the space ensure that we can perform a A. DOW AND Z. FENG further forcing to obtain a model of Martin’s Axiom and that the desired prop-erties of a space naturally generated from the original space possesses these sameproperties in the final model. The reader may be interested to note that in thisway we produce a model of Martin’s Axiom and c = ω in which there is a com-pact space of countable tightness that is not sequential. This is interesting becauseBalogh proved in [4] that the forcing axiom, PFA, implies that compact spacesof countable tightess are sequential. It was first shown in [6] that the celebratedMoore-Mrowka problem was independent of Martin’s Axiom plu c = ω . The meth-ods in [6] are indeed based on the paper [11] using the notion of T-algebras firstformulated in [13]. The example in [11] is itself a space generated by a T-algebrabut is not explicitly formulated as such because of its simpler structure.To do all this, at minimum cost, we must explicitly reference a number of state-ments and proofs from [11]. The construction is modeled on the following naturalproperty of locally compact scattered space with its implicit right-separated or-dering by an ordinal µ . There are functions H with domain µ and a function i : [ µ ] → [ µ ] < ℵ satisfying that for all α < β < µ :(1) α ∈ H ( α ) ⊂ α + 1 and H ( α ) is a compact open set,(2) i ( α, β ) is a finite subset of α ,(3) if α / ∈ H ( β ), then H ( α ) ∩ H ( β ) ⊂ S { H ( ξ ) : ξ ∈ i ( α, β ) } (4) if α ∈ H ( β ), then H ( α ) \ H ( β ) ⊂ S { H ( ξ ) : ξ ∈ i ( α, β ) } .Conversely if H and i are functions as in (1)-(4) where (1) is replaced by simply(1’) α ∈ H ( α ) ⊂ α + 1 (i.e. no mention of topology)then using the family { H ( α ) : α ∈ µ } as a clopen subbase generates a locallycompact scattered topology on µ in which H, i satisfy property (1)-(4).Statements (3) and (4) are combined into a single statement in [11] by adoptingthe notation H ( α ) ∗ H ( β ) = ( H ( α ) ∩ H ( β ) α / ∈ H ( β ) H ( α ) \ H ( β ) α ∈ H ( β ) . As noted in [11] a locally compact scattered space can not have the properties listedabove, hence the construction must be generalized. Also it is shown above (and in[2] for P = ω ω ) that a compact scattered space with a P -base that has Calibre ω will be first countable.The generalization from [11] will use almost the same terminology and ideas togenerate a topology on the base set ω × C where C = 2 N is the usual Cantor setand, for each α < ω , { α } × C will be homeomorphic to C . For n ∈ N = ω \ { } and ǫ ∈
2, the notation [ n, ǫ ] will denote the clopen subset { f ∈ N : f ( n ) = ǫ } in C . However a critically important aspect of the construction to watch for is thatevery point of the space will have a local base of neighborhoods that splits only oneof the sets in {{ α } × C : α ∈ ω } . The function H will identify the copies that sucha subbasic clopen set meets (and contains all except the top one). More precisely, H ( α, × C be a subbasic clopen set, and for n >
0, the set H ( α, n ) ⊂ H ( α, { α } × C as theset { α } × [ n, H ( α, \ H ( α, n ) will generate the subbasic clopenset corresponding to { α } × [ n, i is similarly generalized to be afunction from [ ω ] × ω . Here is the definition of a suitable pair of functions from[11, Definition 2.4]. OMPACT SPACES WITH A P -BASE 9 Definition 5.1.
A pair H : ω × ω → P ( ω ) and i : [ ω ] × ω is ω -suitable if thefollowing conditions hold for all α < β < ω and n ∈ N : (1) α ∈ H ( α, n ) ⊂ H ( α, ⊂ α + 1 , (2) i ( α, β, n ) ∈ [ α ] < ℵ , (3) H ( α, ∗ H ( β, n ) ⊂ S { H ( ξ,
0) : ξ ∈ i ( α, β, n ) } .Also, given an ω -suitable pair H, i , define the following sets for α ∈ ω , F ∈ [ ω ] < ℵ and n ∈ N : (4) U ( α ) = U ( α, C ) = H ( α, × C , (5) U ( α, [ n, { α } × [ n, ∪ (( H ( α, n ) \ { α } ) × C ) , (6) U ( α, [ n, U ( α, C ) \ U ( α, [ n, , (7) U [ F ] = S { U ( ξ ) : ξ ∈ F } . Next we rephrase [11, Lemma 2.5]:
Proposition 5.2. If H, i is an ω -suitable pair then the subbase { U ( α, C ) : α ∈ ω } ∪ { U ( α, [ n, ǫ ]) : α ∈ ω , n ∈ N , ǫ ∈ } generates a locally compact Hausdorff topology τ H on ω × C satisfying that for all α ∈ ω , n ∈ N , and r ∈ C , (1) U ( α, C ) , U ( α, [ n, are compact, (2) the collection of finite intersections of members of the family { U ( α, [ n, r ( n )]) \ U [ F ] : n ∈ N , F ∈ [ α ] < ℵ } is a local base at ( α, r )Next, the authors of [11] have to work very hard to produce an ω -suitable pairso that τ H is first-countable and initially ω -compact. The first step is to work ina model in which there is a special function f : [ ω ] [ ω ] ≤ℵ called a strong∆-function. Since we will not need any properties of this function we omit thedefinition, but henceforth assume that f is such a function. We record additionalminor modifications of results from [11, 4.1,4.2]. Proposition 5.3.
There is a ccc poset P f consisting of quadruples p = ( a p , h p , i p , n p ) that are finite approximations of an ω -suitable pair where (1) a p ∈ [ ω ] < ℵ , n p ∈ ω (2) h p : [ a p ] × n p
7→ P ( a p ) , (3) i p : [ a p ] × n p [ a p ] < ℵ ,and, for each P f -generic filter G , the relations H = [ { h p : p ∈ G } and i = [ { i p : p ∈ G } are functions that form an ω -suitable pair. In particular, if p ∈ G , α ∈ h p ( β ) , and i p ( α, β,
0) = ∅ , then (in V [ G ] ) U ( α, C ) ⊂ U ( β, C ) . The space ( ω × C , τ H ) is shown to have these additional properties [11, 4.2]: Proposition 5.4. If G is P f -generic and H, i are defined as in Proposition 5.3,then the following hold in V [ G ] : (1) X H = ( ω × C , τ H ) is locally compact 0-dimensional of cardinality c = 2 ℵ = ℵ , (2) X H is first-countable, (3) for every A ∈ [ X H ] ω , there is a λ < ω such that A ∩ U ( λ, C ) is uncount-able, (4) for every countable A ⊂ X H , either Y is compact or there is an α < ω such that ( ω \ α ) × C is contained in Y .Consequently X H is a locally compact, -dimensional, normal, first-countable, ini-tially ω -compact but non-compact space. Finally, we need the following strengthening of [11, Lemma 7.1] but which isactually proven.
Proposition 5.5. If p = ( a p , h p , i p , n p ) ∈ P f and a p ⊂ λ ∈ ω , then there is a q < p in P f such, that (1) a q = a p ∪ { λ } and n q = n p , (2) a p ⊂ h q ( λ, , (3) i q ( α, λ, j ) = ∅ for all α ∈ a p and j < n q . We note that for p, q as in Lemma 5.5, if q is in the generic filter G , then U ( α, C ) is a subset of U ( λ, C ) for all α ∈ a p . One consequence of this is that thefamily { U ( α, C ) : α ∈ ω } is finitely upwards directed. Equivalently, the family ofcomplements of these sets in the one-point compactification of X H is a neighborhoodbase for the point at infinity.Now we strengthen [11, Lemma 7.2] which will be used to prove that the one-point compactification of X H has Calibre ω . Some of our proofs will require forcingarguments and we refer the reader to [14] for more details. However some remarksmay be sufficient to assist many readers. The forcing extension, V [ G Q ] by a Q -generic filter G Q for a poset Q is equal to the valuation, val G Q ( ˙ A ) for the collectionof all Q -names ˙ A that are sets from V . The notation q (cid:13) x ∈ ˙ A can be read asthe assertion that x ∈ val G Q ( ˙ A ) for any generic filter with q ∈ G Q . The forcingtheorem ([14, VII 3.6]) ensures, for example, that if ˙ A is a Q -name of a subset ofa ground model set B , then b is an element of val G Q ( ˙ A ) exactly when there is anelement q ∈ G Q such that q (cid:13) b ∈ val G Q ( ˙ A ). Additionally, the set of q ∈ Q thatsatisfy that q (cid:13) x ∈ ˙ A is a set in the ground model, as is the set of x for whichthere exists a q with q (cid:13) x ∈ ˙ A . This justifies the first line of the next proof. Lemma 5.6. In V [ G ] , for each uncountable A ⊂ ω , there is a λ < ω such that U ( α, C ) ⊂ U ( λ, C ) for uncountably many α ∈ A .Proof. Let ˙ A be a P f -name for a subset of ω . Fix any condition p ∈ G and assumethat p forces that ˙ A has cardinality ℵ . We prove that there is a q < p and a λ ∈ a q satisfying that if q ∈ G then there are uncountably many α ∈ val G ( ˙ A ) such that U ( α, C ) ⊂ U ( λ, C ). It is a standard fact of forcing that this would then establishthe Lemma (i.e. that there is then necessarily such a q ∈ G ).Let I denote the set of α ∈ ω satisfying that there is some p α < p (which wechoose) forcing that α ∈ ˙ A . Since p forces that ˙ A is a subset of I it follows that I has cardinality at least ω . Since P f is ccc, it also follows that I has cardinalityequal to ω but it suffices for this argument to choose any λ ∈ ω such that I ∩ λ is uncountable. For each α ∈ I , choose q α < p α so that a q α = a p ∪ { λ } and theproperties of the pair p α , q α are as stated in Proposition 5.5.Just as in the proof of [11, Lemma 7.2], the fact that P f is ccc ensures that thereis some q < p such that so long as q ∈ G , the set { α ∈ I ∩ λ : q α ∈ G } is uncountable. OMPACT SPACES WITH A P -BASE 11 As remarked after Proposition 5.5, it follows, in V [ G ], that U ( α, C ) ⊂ U ( λ, C ) forall α ∈ { α ∈ I ∩ λ : q α ∈ G } . (cid:3) Theorem 5.7. If G is a P f -generic filter, then in V [ G ] , the one-point compactifi-cation of the space X H has a P -base for a poset with Calibre ω .Proof. The poset P consists of the family { U ( α, C ) : α ∈ ω } ordered by inclusion.To complete the proof we have to note that ω < T P . For this it is enough to provethat there is a countable subset of P that has no upper bound. It is relatively easyto prove that X H is separable (indeed, that ω × C is dense) but oddly enough thisis not stated in [11] and we can more easily simply note that X H is not σ -compactbecause by Proposition 5.4 it is countably compact and non-compact. (cid:3) An important feature of the construction of X H from the ω -suitable pair H, i isthat even in a forcing extension by a ccc poset Q (in fact by any poset that preservesthat ω and ω are cardinals), the new interpretation of the space obtained using H, i (i.e. the base set ω × C may change because there can be new elements of C ) is still locally compact and 0-dimensional. This is similar to the fact that localcompactness of scattered spaces is preserved by any forcing (a result by Kunen).The other properties of X H , such as first-countability and initial ω -compactness,as well as properties of its one-point extension are not immediate and will dependon what subsets of ω have been added.An unexpected feature of the ω -suitable pair is that, in fact, the first countabilityof X H is preserved by any forcing. Lemma 5.8.
For each poset Q in V [ G ] and Q -generic filter G Q , the space X H isfirst-countable in V [ G ][ G Q ] .Proof. Of course we will use the fact that, in V [ G ], X H is first-countable (as statedin Proposition 5.4). Fix any α ∈ ω and recall from Proposition 5.2, that thecollection of all finite intersections of the family { U ( α, [ n, r ( n )]) \ U [ F ] : n ∈ N , F ∈ [ α ] < ℵ } is a local base at ( α, r ) ∈ { α } × C (in any model). In V [ G ], for each r ∈ C , let Z ( α, r ) = T n ∈ N U ( α, [ n, r ( n )]) and let K ( α, r ) = { ξ < α : { ξ } × C ⊂ Z ( α, r ) } . Letus recall that ξ ∈ K ( α, r ) if and only if Z ( α, r ) ∩ ( { ξ }× C ) is not empty. Similarly, bythe definition of U ( α, [ n, r ( n )]) given in Definition 5.1, K ( α, r ) = T { H ( α, [ n, r ( n )]) : n ∈ N } . Since, for all n ∈ N , { H ( α, [ n, , H ( α, [ n, } is a partition of H ( α, { K ( α, r ) : r ∈ C } is also a partition of H ( α, X H is first-countable (in V [ G ]), for each r ∈ C , there is a countable F r ⊂ K ( α, r ) such that K ( α, r ) × C ⊂ S { U ( ξ,
0) : ξ ∈ F r } .Now we are ready to show that, in V [ G ][ G Q ], each point of { α } × C is a G δ -pointin X H . For each r ∈ C , we again define the G δ -set Z ( α, r ) and K ( α, r ) ⊂ H ( α,
0) aswe did in V [ G ] but as calculated in the new model V [ G ][ G Q ]. It is immediate that Z ( α, r ) ∩ ( { α } × C ) is equal to ( α, r ). Since there are no changes to the values of H ( α, [ n, ǫ ]) for ( n, ǫ ) ∈ N ×
2, the value of K ( α, r ) for each r ∈ C ∩ V [ G ] is unchangedand the family { K ( α, r ) : r ∈ C } is a partition of H ( α, r ∈ C ∩ V [ G ], K ( α, r ) × C is a subset of S { U ( ξ,
0) : ξ ∈ F r } . Thisimplies that ( α, r ) is a G δ -point for each r ∈ C ∩ V [ G ]. Now consider a point s ∈ C that is not an element of V [ G ]. But now we have that K ( α, s ) is empty since H ( α,
0) is covered by the family { K ( α, r ) : r ∈ C ∩ V [ G ] } . This implies that Z s isequal to the singleton set { ( α, s ) } . (cid:3) Next we prove that we can extend the model V [ G ] to obtain a model in whichMartin’s Axiom holds (and c = ω ). We do so using the following result from[14, VI 7.1, VIII 6.3] (i.e. the standard method to construct a model of Martin’sAxiom). Proposition 5.9.
In the model V [ G ] , there is an increasing chain { Q ξ : ξ ≤ ω } of partially ordered sets satisfying for each ξ < ω (1) Q ξ is a ccc poset of cardinality at most ℵ , (2) each maximal antichain of Q ξ is a maximal antichain of Q ω , (3) if G is a Q ω -generic filter, then in the model V [ G ][ G ](a) Martin’s Axiom holds and c = ω (b) for each A ⊂ ω × C of cardinality less than ω , there is a ξ < ω suchthat A is in the model V [ G ][ G ∩ Q ξ ] . For the remainder of this section let { Q ξ : ξ ≤ ω } be the poset as in thisProposition and let G be a Q ω -generic filter. The model V [ G ][ G ∩ Q ξ ] is actuallyequal to the valuation by G of all Q ξ -names that are in V [ G ].First we prove that the poset of P (consisting of the family { U ( α, C ) : α ∈ ω } ordered by inclusion) still has Calibre ω in the forcing extension of V [ G ] by Q ω .In fact, by Proposition 5.9, it suffices to prove that any ccc poset Q of cardinalityat most ℵ preserves that P has Calibre ω . Lemma 5.10. If G Q is Q -generic over V [ G ] for a ccc poset, then P has Calibre ω in the model V [ G ][ G Q ] .Proof. Let ˙ A be a Q -name of a subset of ω and let q be any element of Q . Let I be the set of α ∈ ω such that there exists some q α < q such that q α (cid:13) α ∈ ˙ A . Foreach α ∈ I choose such a q α < q . Fix any λ < ω so that I λ = { α ∈ I : U ( α, C ) ⊂ U ( λ, C ) } is uncountable. Choose ¯ q < q so that for all Q -generic G Q with ¯ q ∈ G Q ,the set { α ∈ I λ : q α ∈ G Q } is uncountable. Since α ∈ val G Q ( ˙ A ) for all α ∈ I with q α ∈ G Q , this completes the proof that P retains the Calibre ω property. (cid:3) It follows from the results so far that, in the model V [ G ][ G ], the one-pointcompactification of X H has a P -base and that P has Calibre ω . Also, { U ( α,
0) : α ∈ ω } is an open cover of X H that has no countable subcover, so the one-pointcompactification is not first-countable. This completes the proof of the desiredproperties, but it is of independent interest to prove this next result because of theconnection to the Moore-Mrowka problem. Theorem 5.11.
In the model V [ G ][ G ] there is an ω -suitable pair H, i and a poset P of Calibre ω such that each of the following hold: (1) Martin’s Axiom and c = ω , (2) the space X H is locally compact, -dimensional, first-countable, and notcompact, (3) the one-point compactification of X H has a P -base (4) the space X H is initially ω -compact and normal, (5) the one-point compactification of X H is compact, has countable tightness,and is not sequential. OMPACT SPACES WITH A P -BASE 13 Proof.
We have already established items (1), (2), and (3). Item (3) implies that theone-point compactification of X H has countable tightness. Item (5) is an immediateconsequence of items (1)-(4). So it remains to prove item (4). This will require aforcing proof over the model V [ G ]. Before we begin, let us notice that: Fact . In V [ G ], if S is an unbounded subset of ω , then the closure of S × C willcontain ( ω \ α ) × C for some α ∈ ω .This follows from the property in item (4) because of the facts that S × C does nothave compact closure and that the one-point compactification of X H has countabletightness.Recall, from Proposition 5.4, that, in V [ G ], the closure in X H of each countablesubset of X H is either compact or contains ( ω \ α ) × C for some α ∈ ω . Wewill prove that this statement remains true in V [ G ][ G ]. Before doing so we notethat item (4) is a consequence of this claim. It is immediate from (4) that X H iscountably compact. The fact that then X H is initially ω -compact follows from thefact that a compact P -space has no converging ω -sequences. The fact that X H isnormal is noted in [11, §
8] and is similar to the proof that an Ostaszewski space isnormal. Indeed, it follows from item (4) that for any two disjoint closed subsets of X H at least one of them is compact.Let ˙ A be a Q ω -name of a countable subset of X H . Assume there is a q ∈ G such that q forces that the closure of ˙ A is not compact. Note that q forces that forall finite F ⊂ ω , ˙ A \ U [ F ] is not empty. Also, that the closure of ˙ A is forced tomiss { α } × C if and only if ˙ A is forced to miss U ( α, \ U [ F ] for some finite F ⊂ α .By Proposition 5.9, there is a ξ < ω and a Q ξ -name ˙ B satisfying that val G ∩ Q ξ ( ˙ B )is equal to val G ( ˙ A ). By possibly choosing a larger value of ξ , we may assume that q ∈ Q ξ . We first note that it suffices to work with ˙ B and the poset Q ξ . Fact . For each λ ∈ ω , k ∈ N , and t : { , . . . , k } 7→
2, and finite F ⊂ λ , thefollowing are equivalent:(1) val G ( ˙ A ) misses T { U ( λ, [ n, t ( n )]) : 1 ≤ n ≤ k } \ U [ F ],(2) val G ∩ Q ξ ( ˙ B ) misses T { U ( λ, [ n, t ( n )]) : 1 ≤ n ≤ k } \ U [ F ].We must prove that the closure of val G ∩ Q ξ ( ˙ B ) contains { λ } × C for a co-initialset of λ ∈ ω . This means that we are interested in the set of λ ∈ ω such that { λ } × C is not contained in the closure of ˙ B . For any such λ , there must be a q ≥ q λ ∈ Q ξ , an integer k λ and a function t λ : { , . . . , k λ } 7→
2, and a finite F λ ⊂ λ such that q ξ forces that ˙ B is disjoint from T { U ( λ, [ n, t λ ( n )]) : 1 ≤ n ≤ k λ } \ U [ F λ ].Let S denote the set of all λ such that such a sequence h q λ , k λ , t λ , F λ i exists.If α / ∈ S , then q forces that { α } × C is contained in the closure of ˙ B . Ofcourse this also means that q forces that the closure of ˙ B contains the closure of( ω \ S ) × C . In this case, Fact 1 implies that there is an α ∈ ω such that theclosure of ( ω \ S ) × C , and therefore of val G ∩ Q ξ ( ˙ B ) , will contain ( ω \ α ) × C asrequired. Therefore we conclude that if S is unbounded, then ω \ S is bounded.We assume that S is unbounded and obtain a contradiction. Since Q ξ hascardinality ℵ , it follows from the pressing down lemma that there is a stationarysubset S of S , consisting of limits with cofinality ω , and a tuple h ¯ q, k, t, F i suchthat, for all λ ∈ S (1) ¯ q = q λ , k = k λ , t = t λ , and (2) F = F λ .Let W be the union of the family { T { U ( λ, [ n, t ( n )]) : 1 ≤ n ≤ k } \ U [ F ] : λ ∈ S } .Since W is open and the property of item (4) holds in V [ G ], it follows that X H \ W is compact. Choose any finite F ⊂ ω so that X H ⊂ W ∪ U [ F ]. It now followsthat ¯ q forces that ˙ B is contained in U [ F ∪ F ], which is a contradiction. (cid:3) References [1] E.K. van Douwen,
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