aa r X i v : . [ m a t h . GN ] A p r PFA( S )[ S ] and Locally Compact NormalSpaces Franklin D. Tall November 20, 2018
Abstract
We examine locally compact normal spaces in models of formPFA( S )[ S ], in particular characterizing paracompact, countably tightones as those which include no perfect pre-image of ω and in whichall separable closed subspaces are Lindel¨of. We will be using a particular kind of model of set theory constructed with theaid of a supercompact cardinal. These models have been used in [31, 30, 49,47, 46, 48, 53]. We start with a particular kind of Souslin tree — a coherent one — in the ground model. Such trees are obtainable from ♦ [29, 41].Their definition will not concern us here. One then iterates proper partialorders as in the proof of the consistency of PFA (so we need to assume theconsistency of a supercompact cardinal) but only those that preserve theSouslinity of that tree. By [34], that produces a model for PFA ( S ) : PFA restricted to partial orders that preserve S . We then force with S . We shall Research supported by NSERC grant A-7354.2010 AMS Math. Subj. Class. Primary 54A35, 54D15, 54D20, 54D45, 03E35, 03E65;Secondary 54E35.Key words and phrases: PFA( S )[ S ], paracompact, locally compact, normal, perfectpre-image of ω , locally connected, reflection, Axiom R, P-ideal Dichotomy, Dowker space,collectionwise Hausdorff, homogeneous compacta. PFA ( S )[ S ] implies ϕ if ϕ holds whenever we force with S over a modelof PFA( S ), for S a coherent Souslin tree. We shall say ϕ holds in a model ofform PFA ( S )[ S ] if ϕ holds in some particular model obtained this way.PFA( S )[ S ] and particular models of it impose a great deal of structure onlocally compact normal spaces because they entail many useful consequencesof both PFA and V = L . We amalgamate here three previous preprints[45], [47], and [46] dealing with characterizing paracompactness and killingDowker spaces in locally compact normal spaces, as well as with homogeneityin compact hereditarily normal spaces. Our proofs will avoid the difficultset-theoretic arguments in other papers on PFA( S )[ S ] by just quotingthe familiar principles derived there, and so should be accessible to anyset-theoretic topologist.The consequences of PFA( S )[ S ] we shall use and the references in whichthey are proved are: (Balogh’s) PPP (defined below) [19]; ℵ -CWH (locally compact normal spaces are ℵ -collectionwise Hausdorff[49]); PID (P-ideal Dichotomy (defined below) [28]).We also mention for the reader’s interest: MM (compact countably tight spaces are sequential [54]); FCL (every first countable hereditarily Lindel¨of space is hereditarilyseparable [32]); FC ℵ -CWH (every first countable normal space is ℵ -collectionwiseHausdorff [31]); OCA (Open Colouring Axiom [17]); b = ℵ ([29]).In the particular model used in [31], we also have: FCCWH (every first countable normal space is collectionwise Hausdorff[31]);
CWH (locally compact normal spaces are collectionwise Hausdorff [49]);2 xiom R (see below [30]).Proofs of all of these results are published or available in preprints, with theexception of MM , for which only a brief sketch exists. We will thereforeavoid using it.In Section 2 we characterize paracompactness in locally compact normalspaces in certain models of PFA( S )[ S ]. In Section 3, we improve ourcharacterization via the use of P-ideal Dichotomy. In Section 4, we examineapplications of Axiom R. In Section 5 we obtain some reflection results inZFC for (locally) connected spaces. In Section 6, we apply our results toexclude certain locally compact Dowker spaces in models of PFA( S )[ S ]. InSection 7, we apply PFA( S )[ S ] to compact hereditarily normal spaces. Engelking and Lutzer [16] characterized paracompactness in generalizedordered spaces by the absence of closed subspaces homeomorphic tostationary subsets of regular cardinals. This was extended to monotonicallynormal spaces by Balogh and Rudin [8]. Moreover, for first countablegeneralized ordered spaces, one can do better:
Proposition 2.1 [52] . Assuming the consistency of two supercompactcardinals, it is consistent that a first countable generalized ordered spaceis (hereditarily) paracompact if and only if no closed subspace of it ishomeomorphic to a stationary subset of ω . We were interested in consistently obtaining a similar characterization forlocally compact normal spaces. However, as we shall see, the locally compact,separable, normal, first countable, submetrizable, non-paracompact spaceWeiss constructed in [57] has no subspace homeomorphic to a stationarysubset of ω . Nonetheless, for restricted classes of locally compact normalspaces, we can get characterizations of paracompactness that do depend onthe spaces’ relationship with ℵ .We will assume all spaces are Hausdorff.In [30] we proved: 3 emma 2.2. In a particular model (the one of [31]) of form
PFA( S )[ S ] ,every locally compact, hereditarily normal space which does not include aperfect pre-image of ω is paracompact. Lemma 2.2 will follow from what we prove here as well. We can turn thisresult into a characterization as follows.
Theorem 2.3.
There is a model of form
PFA( S )[ S ] in which locally compacthereditarily normal spaces are (hereditarily) paracompact if and only if theydo not include a perfect pre-image of ω . Proof.
The backward direction follows from Lemma 2.2, since a spaceis hereditarily paracompact if every open subspace of it is paracompact,and open subspaces of locally compact spaces are locally compact. The“hereditarily” version of the other direction is because perfect pre-images of ω are countably compact and not compact, and hence not paracompact.Without “hereditarily” we need: Lemma 2.4 [15] . In a countably tight space, perfect pre-images of ω areclosed. Lemma 2.5 [3, 5, 30] . A locally compact space has a countably tight one-pointcompactification if and only if it does not include a perfect pre-image of ω . Note that if X has a countably tight one-point compactification, X itselfis countably tight.We also now have a partial characterization for locally compact spacesthat are only normal: Theorem 2.6.
There is a model of form PFA ( S )[ S ] in which a locallycompact normal space is paracompact and countably tight if and only if itsseparable closed subspaces are Lindel¨of and it does not include a perfectpre-image of ω . The proof of Theorem 2.6 is quite long. It is convenient to first prove theweaker
Theorem 2.7.
There is a model of form
PFA( S )[ S ] in which a locallycompact normal space X is paracompact and countably tight if and only if theclosure of every Lindel¨of subspace of X is Lindel¨of, and X does not includea perfect pre-image of ω . ω willbe excluded by countable tightness plus paracompactness. It is also easyto see that a paracompact space with a dense Lindel¨of subspace is Lindel¨of— since it has countable extent — so closures of Lindel¨of subspaces areLindel¨of. The other direction is harder, but much of the work has been doneelsewhere. We refer to [20] for a definition of the reflection axiom Axiom R . Dow [12] proved it equivalent to stationary set reflection for stationarysubsets of [ κ ] ω , κ uncountable. However, we shall only use the following threeresults concerning it. We have: Lemma 2.8 [30] . Axiom R holds in the PFA ( S )[ S ] model of [31]. Definition. L ( Y ), the Lindel¨of number of Y , is the least cardinal κ suchthat every open cover of Y has a subcover of size ≤ κ . Lemma 2.9 [5] . Axiom R implies that if X is a locally Lindel¨of, regular,countably tight space such that every open Y with L ( Y ) ≤ ℵ has L ( Y ) ≤ ℵ ,then if X is not paracompact, it has a clopen non-paracompact subspace Z with L ( Z ) ≤ ℵ . Lemma 2.10 [5] . Axiom R implies that if X is locally Lindel¨of, regular,countably tight, and not paracompact, then X has an open subspace Y with L ( Y ) ≤ ℵ , such that Y is not paracompact. We also have:
Lemma 2.11. If Y is a subset of a locally Lindel¨of space of countabletightness in which closures of Lindel¨of subspaces are Lindel¨of, then if L ( Y ) ≤ℵ , then L ( Y ) ≤ ℵ . Proof.
For let U be a collection of open sets with Lindel¨of closures covering Y . There are ℵ of them, say { U α } α<ω , which cover Y . Then S α<ω U α = S α<ω S β<α U β = S α<ω S β<α U β ⊇ Y . But each S β<α U β is Lindel¨of, so L (cid:0)S α<ω U α (cid:1) ≤ ℵ . Y is a closed subspace of S α<ω U α , so it too has Lindel¨ofnumber ≤ ℵ .To finish the proof of Theorem 2.7 it therefore suffices to prove: Theorem 2.12.
PFA( S )[ S ] implies that if X is a locally compact normalspace with L ( X ) ≤ ℵ , closures of Lindel¨of subspaces of X are Lindel¨of, and X includes no perfect pre-image of ω , then X is paracompact. ℵ -CWH and: Lemma 2.13 [19], [53] . PFA( S )[ S ] implies PPP : if X is compact and countably tight, and Z ⊆ X is such that | Z | ≤ ℵ and there exists a collection V of open sets, |V| ≤ ℵ , and a collection U = { U V : V ∈ V} of open sets, such that Z ⊆ S V , and for each V ∈ V , there is a U V ∈ U such that V ⊆ V ⊆ U V , and | U V ∩ Z | ≤ ℵ ,then Z is σ -closed discrete in S V . The conclusion of Lemma 2.13 had previously been shown under MA ω by Balogh [3]. The weaker conclusion asserting that Z is σ -discrete, if it’slocally countable, was established by Todorcevic. A modification of his proofyields the stronger result [19]. It follows that: Corollary 2.14.
PFA( S )[ S ] implies that if X is locally compact, includes noperfect pre-image of ω , and L ( X ) ≤ ℵ , and Y ⊆ X , | Y | = ℵ , is such thateach point in X has a neighbourhood meeting at most countably many pointsof Y , then Y is σ -closed-discrete. We now need some results of Nyikos:
Definition.
A space X is of Type I if X = S α<ω X α , where each X α isopen, α < β implies X α ⊆ X β , and each X α is Lindel¨of. { X α : α < ω } is canonical if for limit α , X α = S β<α X β . Lemma 2.15 [38] . If X is locally compact, L ( X ) ≤ ℵ , and every Lindel¨ofsubset of X has Lindel¨of closure, then X is of Type I, with a canonicalsequence. Lemma 2.16 [35] . If X is of Type I, then X is paracompact if and only if { α : X α − X α = 0 } is non-stationary. Proof of Theorem 2.12. If X is paracompact, this is straightforward.Suppose X were not paracompact. X is of Type I so we may pick a canonicalsequence and we may pick a stationary S ⊆ ω and x α ∈ X α − X α , for each α ∈ S . By Corollary 2.14, { x α : α ∈ S } is σ -closed-discrete, so there is astationary set of limit ordinals S ′ ⊆ S such that { x α : α ∈ S ′ } is closeddiscrete. Let { U α : α ∈ S ′ } be a discrete collection of open sets expanding it.Pressing down yields an uncountable closed discrete subspace of some X α ,contradiction. (cid:3) Y of a locally compact, hereditarily normal spacewhich does not include a perfect pre-image of ω . The following argument inNyikos [38] will establish that Y is Lindel¨of. First consider the special casewhen Y is open. Let B be a right-separated subspace of the boundary of Y .We claim B is countable, whence the boundary is hereditarily Lindel¨of, so Y is Lindel¨of. Since the one-point compactification of Y is countably tight, byLemma 2.13, if B is uncountable, it has a discrete subspace D of size ℵ . D isclosed discrete in Z = Y − ( D − D ), so in Z there is a discrete open expansion { U d : d ∈ D } of D , because Y is hereditarily strongly ℵ -collectionwiseHausdorff by CWH . Since Y ⊆ Z , { U d ∩ Y : d ∈ D } is a discrete collectionof non-empty subsets of Y , contradicting Y ’s Lindel¨ofness.Now consider an arbitrary Lindel¨of Y . Since X is locally compact, Y canbe covered by countably many open Lindel¨of sets. The closure of their unionis Lindel¨of and includes Y .We next note that the requirement that Lindel¨of subspaces have Lindel¨ofclosures can be weakened. Recall the following result in [25]: Lemma 2.17.
Every locally compact, metalindel¨of, ℵ -collectionwiseHausdorff, normal space is paracompact. Since in a normal ℵ -collectionwise Hausdorff space the closure ofa Lindel¨of subspace has countable extent, and metalindel¨of spaces withcountable extent are Lindel¨of, we see that it suffices to have that closuresof Lindel¨of subspaces are metalindel¨of. Corollary 2.18.
There is a model of form
PFA( S )[ S ] in which a locallycompact normal space X is paracompact if and only if the closure of everyLindel¨of subset of X is metalindel¨of and X does not include a perfectpre-image of ω . A consequence of Corollary 2.14 is that we can improve Theorem 2.12 forspaces with Lindel¨of number ≤ ℵ to get: Theorem 2.19.
PFA( S )[ S ] implies that if X is a locally compact normalspace with L ( X ) ≤ ℵ , and X includes no perfect pre-image of ω , then X is paracompact. Proof.
As before, it suffices to consider the case of an open Lindel¨of Y . Ifthe closure of Y were not Lindel¨of, since it has Lindel¨of number ≤ ℵ there7ould be a locally countable subspace Z of size ℵ included in Y − Y . Thatsubspace would then be σ -closed-discrete by Corollary 2.14. As in the proofof Lemma 2.2 from Theorem 2.7, we obtain a contradiction by getting anuncountable closed discrete subspace of Y . Since we have σ - closed -discrete,we only need normality rather than hereditary normality.In retrospect, Theorem 2.19 is perhaps not so surprising: a phenomenonfirst evident in [3] is that “normal plus L ≤ ℵ ” can often substitute for“hereditarily normal” in this area of investigation.In fact, an even further weakening is possible: Definition.
Let U be an open cover of a space X and let x ∈ X . Ord (x , U) = |{ U ∈ U : x ∈ U }| . X is submeta- ℵ -Lindel¨of if everyopen cover has a refinement S n<ω U n such that each U n is an open cover, andfor each x ∈ X , there is an n such that | Ord( x, U n ) | ≤ ℵ . Theorem 2.20.
There is a model of form
PFA( S )[ S ] in which a locallycompact normal space is paracompact and countably tight if and only if itis submeta- ℵ -Lindel¨of and does not include a perfect pre-image of ω . Proof.
It suffices to prove that closures of Lindel¨of subspaces have Lindel¨ofnumber ≤ ℵ , for then we can apply Theorem 2.19 to get that closures ofLindel¨of subspaces are Lindel¨of. Thus all we need is Lemma 2.21.
Every submeta- ℵ -Lindel¨of space with extent ≤ ℵ hasLindel¨of number ≤ ℵ . Proof.
Following a similar proof in [4], suppose the space X has an opencover V with no subcover of size ≤ ℵ . Let S n<ω U n be an open refinementof V as in the definition of submeta- ℵ -lindel¨ofness. For each y ∈ X , pick n ( x ) ∈ ω such that | Ord( x, U n ( x ) ) | ≤ ℵ . Let X n = { x ∈ X : n ( x ) = n } .Then for every n < ω , there is a maximal A n ⊆ X n such that no member of U n contains two points of A n . By maximality, V ′ = S n<ω { S { U ∈ U n : x ∈U } : x ∈ A n } covers X n . Since V has no subcover of size ≤ ℵ , | A n | > ℵ forsome n . But | A n | is closed discrete.An immediate corollary of Theorem 2.19 is: Corollary 2.22.
PFA( S )[ S ] implies every locally compact normal space ofsize ≤ ℵ with a G δ -diagonal is metrizable. G δ -diagonal do not admit perfectpre-images of ω , compact spaces with a G δ -diagonal are metrizable, andparacompact locally metrizable spaces are metrizable.Weiss’ space mentioned above constrains attempts at characterizingparacompactness. It is submetrizable, so has a G δ -diagonal. That latterproperty is hereditary; Lutzer [33] proved that linearly ordered spaces witha G δ -diagonal are metrizable, so: Proposition 2.23.
If a space has a G δ -diagonal, it has no subspacehomeomorphic to a stationary set. We are thus going to need stronger constraints on sets of size ℵ thanjust excluding copies of stationary sets, if we wish to weaken the Lindel¨ofrequirement of Theorem 2.7 to just something involving ℵ . Also notethat Weiss’ space prevents us from removing the cardinality restriction fromTheorem 2.19. We will consider some such constraints in Section 4. In order to prove Theorem 2.6, we introduce some known ideas about ideals.
Definition.
A collection I of countable subsets of a set X is a P-ideal ifeach subset of a member of I is in I , finite unions of members of I are in I ,and whenever { I n : n ∈ ω } ⊆ I , there is a J ∈ I such that I n − J is finitefor all n . P (short for P-ideal Dichotomy ): For every P -ideal I on a set X , eitheri) there is an uncountable A ⊆ X such that [ A ] ≤ ω ⊆ I or ii) X = S n<ω B n such that for each n , B n ∩ I is finite, for all I ∈ I . Definition. [15] An ideal I of subsets of a set X is countable-covering iffor each countable Q ⊆ X , there is { I Qn : n ∈ ω } ⊆ I , such that for each I ∈ I such that I ⊆ Q , I ⊆ I Qn for some n . CC : For every countable-covering ideal I on a set X , either:i) there is an uncountable A ⊆ X such that A ∩ I is finite, for all I ∈ I ,or ii) X = S n<ω B n such that for each n , [ B n ] ≤ ω ⊆ I .9odorcevic’s proof that P F A ( S )[ S ] implies P appears in [28]. In [15],Eisworth and Nyikos proved that P implies CC , and also proved the followingremarkable result: Lemma 3.1. CC implies that if X is a locally compact space, then eithera) X is the union of countably many ω -bounded subspaces,or b) X does not have countable extent,or c) X has a separable closed subspace which is not Lindel¨of. Recall a space is ω -bounded if every countable subspace has compactclosure. ω -bounded spaces are obviously countably compact.From [23] we have: Lemma 3.2. An ω -bounded space is either compact or includes a perfectpre-image of ω . We can now prove Theorem 2.6.The forward direction follows from 2.7. To prove the other direction, itsuffices to show that if Y is a Lindel¨of subspace of our space X , then Y isLindel¨of. Applying 3.1, we see that by 3.2, Y will be σ -compact if we canexclude alternatives b) and c). c) is excluded by hypothesis, so it suffices toshow that Y has countable extent. But that is easily established, since Y is locally compact normal and hence ℵ -CWH . A closed discrete subspaceof size ℵ in Y could thus be fattened to a discrete collection of open sets.Their traces in Y would contradict its Lindel¨ofness. . Corollary 3.3.
There is a model of form
PFA( S )[ S ] in which a locallycompact space is metrizable if and only if it is normal, has a G δ -diagonal,and every separable closed subspace is Lindel¨of. Proof.
Theorem 2.7 applies, since spaces with G δ -diagonals do not includeperfect pre-images of ω .This characterization does not hold in ZFC; the tree topology on aspecial Aronszajn tree is a locally compact Moore space, and hence has a G δ -diagonal. Under MA ω , it is (hereditarily) normal. See e.g. the surveyarticle [51]. Every separable subspace of an ω -tree is bounded in height, andso is countable. 10he condition in Theorem 2.6 that separable closed sets are Lindel¨of, i.e.countable sets have Lindel¨of closures, can actually be weakened by a differentargument, although perhaps the proof is more interesting than the result. Theorem 3.4.
There is a model of form
PFA( S )[ S ] is which a locallycompact normal space is paracompact and countably tight if and only if itincludes no perfect pre-image of ω and the closure of each countable discretesubspace is Lindel¨of. We need several auxiliary results before proving this.
Lemma 3.5 [2] . If X is Tychonoff, countably tight, ℵ -Lindel¨of, andcountable discrete sets have Lindel¨of closures, then X is Lindel¨of. Recall a space is defined to be ℵ -Lindel¨of if every open cover of size ℵ has a countable subcover; equivalently, if every set of size ℵ has a completeaccumulation point. Theorem 3.6.
Assume
PFA( S )[ S ] . Let X be locally compact, normal, andnot include a perfect pre-image of ω . Then either:a) X is σ -compact,or b) e ( X ) > ℵ ,or c) X has a countable discrete subspace D such that D is not Lindel¨of. Proof.
Assume b) and c) fail. Since X is locally compact and countablytight, by Lemma 3.5, it suffices to prove X is ℵ -Lindel¨of.If not, there is a Y ⊆ X of size ℵ with no complete accumulationpoint. Thus Y is locally countable and hence σ -discrete. Hence there isan uncountable discrete Z ⊆ Y with no complete accumulation point. Let Z = { z α : α < ω } . Then Z = S β<ω { z α : α < β } . By hypothesis, itfollows that L ( Z ) ≤ ℵ . But then Z is paracompact by Theorem 2.7. But e ( X ) ≤ ℵ , so Z is Lindel¨of, so Z does have a complete accumulation point,giving a contradiction. Proof of Theorem 3.4.
It suffices to show closures of countablesubspaces of our spaces are Lindel¨of. By normality and ℵ -collectionwiseHausdorfness they have countable extent, and we are assuming countablediscrete subspaces have Lindel¨of closures, so we are done by Theorem 3.6 (cid:3) One is tempted to substitute the condition that discrete subspaces haveLindel¨of closures for b) and c) of Theorem 3.6, but unfortunately it is alreadyknown that countably tight spaces with that property are Lindel¨of [2].11
Applications of Axiom R and HereditaryParacompactness
We shall consider hereditary paracompactness and obtain some interestingresults. We will need the following result of Balogh [6]:
Lemma 4.1. If X is countably tight, has a dense subspace of size ≤ ℵ ,and every subspace of size ≤ ℵ is metalindel¨of, then X is hereditarilymetalindel¨of. Balogh assumes in [6] that all spaces considered are regular, but doesnot use regularity in the proof of Lemma 4.1. He also does not actuallyrequire all subspaces of size ≤ ℵ to be metalindel¨of in order to obtain theconclusion of Lemma 4.1. We refer the reader to [6] for the details. Notethat it follows that in a countably tight space in which every subspace of size ≤ ℵ is metalindel¨of, separable sets are (hereditarily) Lindel¨of. Also notethat Weiss’ space must have a subspace of size ℵ which is not metalindel¨of. Theorem 4.2.
Axiom R implies a locally separable, regular, countably tightspace is hereditarily paracompact if and only if every subspace of size ≤ ℵ ismetalindel¨of. Proof.
One direction is trivial. To go the other way, we shall first obtainparacompactness via Lemma 2.10. Here we do need regularity. I thank SakaeFuchino for pointing this out. Let V be an open subspace with L ( V ) ≤ ℵ .Covering V by ≤ ℵ separable open sets, we see that d ( V ) ≤ ℵ . Thenby Lemma 4.1, V is hereditarily paracompact. To get the whole spacehereditarily paracompact, note it is a sum of separable, hence hereditarilyLindel¨of, clopen sets. Corollary 4.3.
Axiom R implies that a locally hereditarily separable, regularspace is hereditarily paracompact if and only if each subspace of size ≤ ℵ ismetalindel¨of. Proof.
Local hereditary separability implies countable tightness.
Corollary 4.4.
Axiom R implies a locally second countable, regular space ismetrizable if and only if every subspace of size ≤ ℵ is metalindel¨of. roof. This is clear, since such a space is locally hereditarily separable,while paracompact, locally metrizable spaces are metrizable.
Corollary 4.5 [5] . Axiom R implies every locally compact space in whichevery subspace of size ℵ has a point-countable base is metrizable. Proof.
Dow [11] showed that compact spaces in which every subspace ofsize ℵ has a point-countable base are metrizable. Corollary 4.6.
Axiom R implies a locally compact space is metrizable if andonly if it has a G δ -diagonal and every subspace of size ≤ ℵ is metalindel¨of. Proof.
Compact spaces with G δ -diagonals are metrizable. Note:
Results similar to ours concerning Axiom R were obtained by S.Fuchino and his collaborators independently [21], [22].With the added power of PFA( S )[ S ], we can utilize Lemma 4.1 withoutassuming local separability. First, we observe: Theorem 4.7. If X is countably tight and every subspace of size ≤ ℵ ismetalindel¨of, then X does not include a perfect pre-image of ω . Proof of Theorem 4.7.
This follows immediately from Lemma 4.1 and:
Lemma 4.8.
Every perfect pre-image of ω includes one of density ≤ ℵ . Proof.
Let π : X → ω be perfect and onto. Let C = { α : π − ( α + 1) − π − ( α ) = 0 } . Then C is unbounded, for suppose not. Then there is an α such that Y = π − ( α + 1). But then Y is compact, contradiction. Pick foreach α ∈ C , a d α ∈ π − ( α + 1) − π − ( α ). Let Q = { d α : α ∈ C } . Then π | Q is perfect, so π ( Q ) is closed unbounded, so is homeomorphic to ω .From Theorem 4.7 we then obtain: Theorem 4.9.
In the model of form
PFA( S )[ S ] of [31] a locally compact,normal, countably tight space is paracompact if every subspace of size ℵ ismetalindel¨of. Proof.
Separable subspaces are Lindel¨of; by 4.8 and 4.1, there are no perfectpre-images of ω in such spaces. 13 orollary 4.10. In the
PFA( S )[ S ] model of [31], a locally compact,countably tight space is hereditarily paracompact if and only if it is hereditarilynormal and every subspace of size ≤ ℵ is metalindel¨of. Proof.
This follows from Theorem 4.7 and the observation that countabletightness is inherited by open subspaces.
Corollary 4.11.
There is a model of form
PFA( S )[ S ] in which a locallycompact, locally separable space is hereditarily paracompact if and only if itis hereditarily normal and every subspace of size ≤ ℵ is metalindel¨of. This will follow from Theorem 4.10 and FC ℵ -CWH , since the latterimplies the hypothesis of the following: Lemma 4.12 [36] . If separable, normal, first countable spaces do not haveuncountable closed discrete subspaces, then compact, separable, hereditarilynormal spaces are countably tight.
We can avoid introducing the hitherto unused axiom FC ℵ -CWH byquoting: Lemma 4.13 [50] . If there is a separable, normal, first countable space withan uncountable closed discrete subspace, there is a locally compact one.
Proof of Corollary 4.11.
It suffices to show such a space is countablytight. Given x ∈ Y , there is a separable open neighbourhood U of x with U compact. Then x ∈ ( U ∩ Y ). U is countably tight by Lemma 4.12. Thusthere is a countable D ⊆ U ∩ Y such that x ∈ D ∩ U . But then x ∈ D asrequired. (cid:3) There is another way of proving Corollary 4.11, which actually givesa slightly stronger result: locally satisfying the countable chain conditioninstead of locally separable. This follows from [53], in which Todorcevicshowed that
PFA implies compact, hereditarily normal spaces satisfyingthe countable chain condition are hereditarily Lindel¨of (and hence firstcountable). Since [53] is still unavailable, we shall prove this in Section 6.14 (Local)Connectedness and ZFC Reflections
One can sometimes replace our use of Axiom R by the assumption of (local)connectedness, thanks to the following observation:
Lemma 5.1 [15, 5.9] . If X is locally compact, locally connected, andcountably tight, then X is a topological sum of Type I spaces if and only ifevery Lindel¨of subspace of X has Lindel¨of closure. Similarly, if X is locallycompact, connected, countably tight, and Lindel¨of subspaces have Lindel¨ofclosures, then it is Type I. Thus we have:
Theorem 2.12 ′ . PFA( S )[ S ] implies a locally compact, locally connected,normal space X is paracompact if and only if separable closed subspaces areLindel¨of, and X does not include a perfect pre-image of ω . Theorem 4.2 ′ . A locally compact, (locally) connected, locally separable,countably tight, regular space is hereditarily paracompact if and only if everysubspace of size ≤ ℵ is metalindel¨of. Proofs.
Theorem 4.2 ′ is the only one which requires a bit of thought. AnyLindel¨of subspace is included in a separable open set S . S is Lindel¨of andtherefore so is L . Thus the space is a sum of Type I spaces, each of density ≤ ℵ , and by Lemma 4.1, each of these is hereditarily metalindel¨of. By localseparability, the space is then hereditarily paracompact. Corollary 4.3 ′ . A locally compact, (locally) connected, locally hereditarilyseparable, regular space is hereditarily paracompact if and only if eachsubspace of size ≤ ℵ is metalindel¨of. Particularly pleasant is:
Corollary 5.2.
A manifold is metrizable if and only if every subspace of size ℵ is metalindel¨of. Corollary 4.5 ′ . A locally compact, (locally) connected space in which everysubspace of size ℵ has a point-countable base is metrizable. Corollary 4.6 ′ . A locally compact, (locally) connected space is metrizable ifand only if it has a G δ -diagonal and every subspace of size ℵ is metalindel¨of.
15e also have:
Theorem 4.7 ′ . PFA( S )[ S ] implies a locally compact, normal, countablytight, connected or locally connected space is paracompact if every subspaceof size ℵ is metalindel¨of. Theorem 4.9 ′ . PFA( S )[ S ] implies a locally compact, locally connected,countably tight space is hereditarily paracompact if and only if it is hereditarilynormal and every subspace of size ≤ ℵ is metalindel¨of. Corollary 4.12 ′ . PFA( S )[ S ] implies a locally compact, locally connected,locally separable space is hereditarily paracompact if and only if it ishereditarily normal and every subspace of size ≤ ℵ is metalindel¨of. Balogh [5] proved:
Lemma 5.3.
Let X be a locally Lindel¨of, regular, countably tight spacewith L ( X ) ≤ ℵ . Suppose that every subspace of size ≤ ℵ of X isparacompact, and X is either normal or locally has countable spread. Then X is paracompact. We then have the following variation of Corollary 4.3 ′ : Theorem 5.4.
Let X be a locally compact, (locally) connected space in whichevery subspace of size ≤ ℵ is metalindel¨of, and which locally has countablespread. Then X is hereditarily paracompact. Proof.
It suffices to show X is paracompact, since all the properties inquestion are open-hereditary. By Lemmas 5.1 and 5.3, it suffices to provethat X is countably tight and closures of Lindel¨of subspaces are Lindel¨of.Lindel¨of subspaces are included in the union of countably many subspaceswith countable spread and hence have countable spread. If a Lindel¨of Y ⊆ X did not have (hereditarily) Lindel¨of closure, there would be a right-separatedsubset Z of Y , with | Z | = ℵ . But Z would then be metalindel¨of and locallycountable, hence paracompact and σ -discrete. Note that X — and hence Z — is countably tight, since compact spaces with countable spread arecountably tight [1]. By Lemma 4.1, Z is then hereditarily metalindel¨of. Z islocally separable, since if U is an open subspace of X with countable spread, Z ∩ U is dense in Z ∩ U , but is countable, since Z is σ -discrete. Similarly theclosure of Z in Y is locally separable. But the closure of Z in Y is Lindel¨of, soit’s separable. But then Z is separable. But then Z is hereditarily Lindel¨of,contradiction. 16he advantage of eliminating explicit and implicit uses of Axiom R as wedid in 2.12 ′ and 4.12 ′ is that it makes it likely that such results can then beobtained without the necessity of assuming large cardinals, by using ℵ -p.i.c.forcing as in e.g. [55]. ( S )[ S ] and Locally Compact DowkerSpaces The question of whether there exist small Dowker spaces , i.e. normal spaceswith product with the unit interval not normal, which have familiar cardinalinvariants of size ≤ ℵ , continues to attract attention from set-theoretictopologists. See for example the surveys [7, 40, 42, 44]. Although there aremany consistent examples, there have been very few results asserting theconsistency of the non-existence of such examples. We shall partially remedythat situation here. In this section, we observe that PFA( S )[ S ] excludes somepossible candidates for small Dowker spaces . Most of our results follow easilyfrom what we have already proved. Recall: Lemma 6.1 [13] . For a normal space X , the following are equivalent:a) X is countably paracompact,b) X × [0 , is normal,c) X × ( ω + 1) is normal. “Small” is not very well-defined; in the recent survey [42], Szeptyckiconcentrates on the properties cardinality ℵ , first countability , separability , local compactness , local countability (i.e. each point has a countableneighbourhood) and submetrizability (i.e. the space has a weaker metrizabletopology). We shall deal with several of these, weakening — in terms ofnon-existence — cardinality ≤ ℵ to Lindel¨of number ≤ ℵ and submetrizable to not including a perfect pre-image of ω . Note that submetrizable spaceshave G δ -diagonals and hence cannot include perfect pre-images of ω , sincecountably compact spaces with G δ -diagonals are metrizable [9]. Main Theorem PFA( S )[ S ] implies there is no locally compact, hereditarily normalDowker space which in addition: ) satisfies the countable chain condition,or b) includes no perfect pre-image of ω and is either connected orlocally connected.or c) has countable extent.2) PFA( S )[ S ] implies there is no locally compact Dowker space whichincludes no perfect pre-image of ω and has Lindel¨of number ≤ ℵ .3) In the PFA ( S )[ S ] model of [31]:(a) there is no locally compact, hereditarily normal Dowker spaceincluding a perfect pre-image of ω .(b) there is no locally compact Dowker space in which separable closedsubspaces are Lindel¨of and which includes no perfect pre-image of ω .(c) there is no locally compact, countably tight Dowker space in whichevery subspace of size ℵ is metalindel¨of.(d) there is no locally compact, countably tight, Dowker D -space. We shall start with:
Theorem 6.2.
Assume
PFA( S )[ S ] . Let X be a locally compact, hereditarilynormal space satisfying the countable chain condition. Then X is hereditarilyLindel¨of, and hence countably paracompact. Proof.
This follows from [53], where Todorcevic proves:
Lemma 6.3.
PFA( S )[ S ] implies compact hereditarily normal spacessatisfying the countable chain condition are hereditarily Lindel¨of. Proof.
Since open subspaces are locally compact normal, the space ishereditarily ℵ -collectionwise-Hausdorff and hence has countable spread. Ifthe space were not hereditarily Lindel¨of, it would have an uncountableright-separated subspace, and hence, by PPP , an uncountable discretesubspace, contradiction.Todorcevic’s proof was more difficult, since ℵ -CWH was not availableto him. 18he one-point compactification of a locally compact, hereditarily normalspace X is hereditarily normal, and satisfies the countable chain conditionif and only if X does. The result follows, so we have established 1a) of theMain Theorem.1b) of the Main Theorem follows from 2.19 plus 5.1.To prove 1c), we call on Theorem 2.6. Since separable closed subspacesare Lindel¨of, the space is the union of countably many ω -bounded – hencecountably compact – subspaces. In a normal space, the closure of a countablycompact subspace is countably compact, and it is not hard to show that theunion of countably many countably compact closed subspaces of a normalspace is countably paracompact.Restating 2) of the Main Theorem, we next have: Theorem 6.4.
PFA( S )[ S ] implies every locally compact Dowker space ofLindel¨of number ≤ ℵ includes a perfect pre-image of ω . Corollary 6.5.
PFA ( S )[ S ] implies there are no locally compact submetrizableDowker spaces of size ℵ . Proof. ω that locally compact spaces of size ℵ which don’tinclude a perfect pre-image of ω are σ -closed-discrete, hence, if normal,are countably paracompact.To show that Theorem 6.2 is not vacuous, we note that Nyikos [37]constructed, assuming ♦ , a hereditarily separable, locally compact, firstcountable, hereditarily normal Dowker space.In [26], the authors remark that they can construct under ♦ , usingtheir technique of refining the topology on a subspace of the real line, alocally compact Dowker space. By CH, such a space has cardinality ℵ .Since it refines the topology on a subspace of R , it is submetrizable. Thusthe conclusion of Corollary 6.5 is independent. We do not have consistentcounterexamples for clauses 1b), 3b), c), d) of the Main Theorem.Clause 3a) of the Main Theorem follows immediately from 2.2; 3b) followsfrom 2.6. To prove 3c), first observe that by 4.7, X does not include a perfectpre-image of ω . Next, if Y is a separable closed subspace of X , by 4.1 Y isLindel¨of.“ D -spaces” are popular these days. See e.g. [14], [24].19 efinition. X is a D -space if for every neighborhood assignment { V x } x ∈ X ,there is a closed discrete Y ⊆ X such that S { V x : x ∈ Y } is a cover. Theorem 6.6.
There is a model of form
PFA( S )[ S ] in which a locallycompact normal countably tight space is paracompact if and only if it is a D -space. Clause 3d) of the Main Theorem follows. Theorem 6.6 is analogous tothe fact that linearly ordered spaces are paracompact if and only if they are D -spaces (see e.g. [24]). Proof of Theorem 6.6.
Assume the space is D . It is well-known and easyto see that countably compact D -spaces are compact. It is also easy to seethat closed subspaces of D -spaces are D .It follows from Lemma 2.4 that a countably tight D -space cannot includea perfect pre-image of ω . By ℵ -CWH , the closure of a countable subspaceof our space is collectionwise Hausdorff, and hence has countable extent. Butagain, it is well-known that D -spaces with countable extent are Lindel¨of. ByTheorem 2.6, our space is then paracompact.For the other direction, a paracompact, locally compact space is a discretesum of σ -compact spaces. It is well-known that σ -compact spaces are D -spaces, and it is easy to verify that discrete sums of D -spaces are D -spaces. (cid:3) One way of strengthening normality without necessarily implyingcountable paracompactness is to assume hereditary normality. Another isto assume powers of the space are normal. And then one could assume both.Let’s see what happens. We have already looked at hereditary normality;but let us also recall from [31] that:
Proposition 6.7.
There is a model of form
PFA( S )[ S ] in which every locallycompact space with hereditarily normal square is metrizable. Even in ZFC, a hereditarily normal square has consequences. Thefollowing results are due to P. Szeptycki [43]:
Proposition 6.8. If X is normal and X includes a countable non-discretesubspace, then X is countably paracompact. Corollary 6.9. If X is separable, first countable, or locally compact, and X is normal, then X is countably paracompact.
20n the other hand, following a suggestion of W. Weiss, Szeptycki [43]noticed that Rudin’s ZFC Dowker space [39] has all finite products normal.Although our consistency results concerning small Dowker spaces improveprevious ones, they have two unsatisfactory aspects. First of all, all but 1c)prove paracompactness, rather than countable paracompactness, so thereought to be sharper results.It is likely that in our results involving hereditary normality, “perfectpre-image of ω ” can be weakened to “copy of ω .” This would follow fromthe following conjecture and unpublished theorem of the author. Conjecture.
PFA( S )[ S ] implies every first countable perfect pre-image of ω includes a copy of ω . Theorem 6.10.
PFA( S )[ S ] implies that every hereditarily normal perfectpre-image of ω includes a first countable perfect pre-image of ω . Another unsatisfactory aspect of our consistency results is that asupercompact cardinal is required to construct models of form PFA( S )[ S ].This is surely overkill, when we are really concerned with ℵ . We suspect thatlarge cardinals are not needed except possibly for those relying on Axiom R.The other clauses probably can be obtained without any large cardinals, by ℵ -p.i.c. forcing as in e.g. [55]. Under PFA( S )[ S ], hereditarily normal compact spaces – “ T compacta” forshort – have strong properties. We have already seen (6.2) that separableones are hereditarily Lindel¨of. It follows that they are first countable. Hence: Theorem 7.1.
Countably compact, locally compact T spaces are sequentiallycompact. Corollary 7.2.
PFA ( S )[ S ] implies T compacta are sequentially compact. Proof of Theorem 7.1.
Let X be a countably compact, locally compact T space. The one-point compactification of the closure of the range of asequence is a separable T compactum, so is first countable. The closure ofthe range is then itself first countable, so there is a subsequence convergingto a limit point of the range. (cid:3) Lemma 7.3. T compacta which are homogeneous and hereditarily strongly ℵ -collectionwise-Hausdorff are countably tight. It follows by ℵ -CWH that PFA( S )[ S ] implies homogeneous T compactaare countably tight. But we can do better: Theorem 7.4.
PFA ( S )[ S ] implies homogeneous T compacta are firstcountable. Proof.
The authors of [27] show that homogeneous T compacta are firstcountable, provided their open Lindel¨of subspaces have hereditarily Lindel¨ofboundaries. We proved this following the proof of 2.12 above.The conclusion of 7.4 was earlier proved consistent by de la Vega, usinga different model [10].The conclusion of 7.2 is not true in ZFC: Fedorchuk’s S -space from ♦ [18] is a T compactum which is countably tight – because it is hereditarilyseparable – but has no non-trivial convergent sequences. Remark.
The proofs in this paper were produced around 2006-2007,assuming ℵ -CWH for the PFA( S )[ S ] ones. However, a correct proof ofthat was only obtained in 2010. At the 2006 Prague Topological Symposium,Todorcevic announced the σ -discrete version of PPP followed from PFA( S )[ S ].Larson [28] wrote some notes on Todorcevic’s lectures at the conference on Advances in Set-theoretic Topology, in Honor of T. Nogura in Erice, Italy in2008 [54]. Using these and ideas of Todorcevic, A. Fischer and the authorderived a proof of
PPP from PFA( S )[ S ] [19]. Acknowledgement.
I am grateful to members of the Toronto SetTheory Seminar and to Gary Gruenhage and Sakae Fuchino for discussionsconcerning this work. I also thank the referee of [47] for correcting thestatements of the D -space results, and for suggesting I merge [47] and [45]. References [1]
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