On the hereditary paracompactness of locally compact, hereditarily normal spaces
aa r X i v : . [ m a t h . GN ] A p r ON THE HEREDITARY PARACOMPACTNESS OFLOCALLY COMPACT, HEREDITARILY NORMAL SPACES
Paul Larson and Franklin D. Tall November 29, 2010
Abstract.
We establish that if it is consistent that there is a supercom-pact cardinal, then it is consistent that every locally compact, hereditarilynormal space which does not include a perfect pre-image of ω is heredi-tarily paracompact. This is the fifth in a series of papers ([LTo], [L ], [FTT], [LT], [T ]being the logically previous ones) that establish powerful topological con-sequences in models of set theory obtained by starting with a particularkind of Souslin tree S , iterating partial orders that don’t destroy S , andthen forcing with S . The particular case of the theorem stated in the ab-stract when X is perfectly normal (and hence has no perfect pre-imageof ω ) was proved in [LT], using essentially that locally compact perfectlynormal spaces are locally hereditarily Lindel¨of and first countable. Here weavoid these two last properties by combining the methods of [B ] and [T ].To apply [B ], we establish the new set-theoretic result that PFA ++ ( S )[ S ]implies Fleissner’s “Axiom R”. This notation is explained below; the modelis a strengthening of those used in the previous four papers. AMS Subj. Class. (2010): Primary 54D35, 54D15, 54D20, 54D45, 03E65; Secondary03E35.
Key words and phrases. locally compact, hereditarily normal, paracompact, AxiomR, PFA ++ . The first author acknowledges support from Centre de Recerca Mathem`atica andfrom NSF-DMS-0801009. The second author acknowledges support from NSERC grant A-7354. PAUL LARSON AND FRANKLIN D. TALL The results established here were actually proved around 2004, moduloresults of Todorcevic announced in 2002 (which now appear in [FTT] and[L ]) and of the second author [T ]. We have delayed submission until acorrect version of [T ] existed in preprint form. Definition.
A continuous map is perfect if images of closed sets areclosed, and pre-images of points are compact.
It is easy to find locally compact, hereditarily normal spaces which arenot paracompact – ω is one such. Non-trivial perfect pre-images of ω mayalso be hereditarily normal, but are not paracompact. Our result says thatconsistently, any example must in fact include such a canonical example. Theorem 1.
If it is consistent that there is a supercompact cardinal, it’sconsistent that every locally compact, hereditarily normal space that doesnot include a perfect pre-image of ω is (hereditarily) paracompact. This is not a ZFC result, since there are many consistent examples oflocally compact, perfectly normal spaces which are not paracompact. Forexample, the Cantor tree over a Q -set, which is the standard example of alocally compact, normal, non-metrizable Moore space – see e.g. [T ], whichhas essentially the same example. Other examples include the Ostaszewskiand Kunen lines, as in [FH].Let us state some axioms we will be using. PFA ++ : Suppose P is a proper partial order, { D α } α<ω is a collection ofdense subsets of P , and { ˙ S α : α < ω } is a sequence of terms such that ( ∀ α < ω ) (cid:13) P ˙ S α is stationary in ω . Then there is a filter G ⊆ P suchthat(i) ( ∀ α < ω ) G ∩ D α = 0 ,and (ii) ( ∀ α < ω ) S α ( G ) = { ξ < ω : ( ∃ p ∈ G ) p (cid:13) ξ ∈ ˙ S α } is stationary in ω . Baumgartner [Ba] introduced this axiom and called it “PFA + ”. Sincethen, others have called this “PFA ++ ”, using “PFA + ” for the weaker one-term version. As Baumgartner observed, the usual consistency proof for AUL LARSON AND FRANKLIN D. TALL PFA, which uses a supercompact cardinal, yields a model for what we arecalling PFA ++ . Definition. Γ ⊆ [ X ] <κ is tight if whenever { C α : α < δ } is an increasingsequence from Γ , and ω < cf δ < κ , then S { C α : α < δ } ∈ Γ . Axiom R: if Σ ⊆ [ X ] <ω is stationary and Γ ⊆ [ X ] <ω is tight and cofinal, then thereis a Y ∈ Γ such that P ( Y ) ∩ Σ is stationary in [ Y ] <ω . Axiom R ++ : if Σ α ( α < ω ) are stationary subsets of [ X ] <ω and Γ ⊆ [ X ] <ω is tight andcofinal, then there is a Y ∈ Γ such that P ( Y ) ∩ Σ α is stationary in [ Y ] <ω for each α < ω . Fleissner introduced Axiom R in [Fl] and showed it held in the usualmodel for PFA.Σ + Σ + Σ + : Suppose X is a countably tight compact space, L = { L α } α<ω a col-lection of disjoint compact sets such that each L α has a neighborhood thatmeets only countably many L β ’s, and V is a family of ≤ ℵ open subsetsof X such that:a) S L ⊆ S V b) For every V ∈ V there is an open U V such that V ⊆ U V and U V meets only countably many members of L .Then L = S n<ω L n , where each L n is a discrete collection in S V . Balogh [B ] proved that MA ω implies the restricted version of Σ + inwhich we take the L α ’s to be points. We will call that “Σ ′ ”. Definition.
A space is (strongly) κ -collectionwise Hausdorff if for eachclosed discrete subspace { x d } d ∈ D , | D | ≤ κ , there is a disjoint (discrete)family of open sets { U d } d ∈ D with x d ∈ U d . A space is (strongly) collection-wise Hausdorff if it is (strongly) κ -collectionwise Hausdorff for all κ . It is easy to see that normal ( κ − ) collectionwise Hausdorff spaces arestrongly ( κ − ) collectionwise Hausdorff.Balogh [B ] proved: PAUL LARSON AND FRANKLIN D. TALL Lemma 2. MA ω + Axiom R implies locally compact hereditarily strongly ℵ -collectionwise Hausdorff spaces which do not include a perfect pre-imageof ω are paracompact. The consequences of MA ω he used are Σ ′ and Szentmikl´ossy’s result [S]that compact spaces with no uncountable discrete subspaces are hereditarilyLindel¨of . Our plan is to find a model in which these two consequences andAxiom R hold, as well as normality implying (strongly) ℵ -collectionwiseHausdorffness for the spaces under consideration. The model we will con-sider is of the same genre as those in [LTo], [L ], [FTT], [LT], and [T ]. Onestarts off with a particular kind of Souslin tree S , a coherent one, which isobtainable from ♦ or by adding a Cohen real. One then iterates in stan-dard fashion as in establishing MA ω or PFA, but omitting partial ordersthat adjoin uncountable antichains to S . In the PFA case for example,this will establish PFA(S) , which is like PFA except restricted to partialorders that don’t kill S . In fact it will also establish PFA ++ (S), whichis the corresponding modification of PFA ++ . We then force with S . Formore information on such models, see [Mi] and [L ]. We use PFA ++ ( S )[ S ] implies ϕ to mean that whenever we force over a model of PFA ++ ( S ) with S , ϕ holds. Similarly for PFA( S )[ S ], etc.In [T ] it is established that: Lemma 3.
PFA(S)[S] implies that locally compact normal spaces are ℵ -collectionwise Hausdorff. By doing some preliminary forcing (as in [LT]), one can actually get fullcollectionwise Hausdorffness, but we won’t need that here.We will assume all spaces are Hausdorff, and use “ X ∗ ” to refer to theone-point compactification of a locally compact space X .There is a bit of a gap in Balogh’s proof of Lemma 2. Balogh assertedthat: Lemma 4. If X is locally compact and does not include a perfect pre-imageof ω , then X ∗ is countably tight. and referred to [B ] for the proof. However in [B ], he only proved thisfor the case in which X is countably tight. It is not obvious that that AUL LARSON AND FRANKLIN D. TALL hypothesis can be omitted, but in fact it can. We need a definition andlemma. Definition.
A space Y is ωωω -bounded if each separable subspace of Y hascompact closure. Lemma 5. [G] , [Bu] . If Y is ω -bounded and does not include a perfectpre-image of ω , then Y is compact. We then can establish Lemma 4 as follows.
Proof.
By Lemma 5, every ω -bounded subspace of X is compact. By [B ],it suffices to show X is countably tight. Suppose, on the contrary, that thereis a Y ⊆ X which is not closed, but is such that for all countable Z ⊆ Y , Z ⊆ Y . Since X is a k -space, there is a compact K such that K ∩ Y is notclosed. Then K ∩ Y is not ω -bounded, so there is a countable Z ⊆ K ∩ Y such that Z ∩ K ∩ Y is not compact. But Z ⊆ Y , so Z ∩ K ∩ Y = Z ∩ K ,which is compact, contradiction.Lemma 3 takes care of the hereditary strong ℵ -collectionwise Haus-dorffness we need, since if open subspaces are ℵ -collectionwise Hausdorff,all subspaces are, and open subspaces of locally compact spaces are locallycompact. The proposition thatΣΣΣ: in a compact countably tight space, locally countable subspaces of size ℵ are σ -discrete. is implied by PFA( S )[ S ] was announced by Todorcevic in the Toronto SetTheory Seminar in 2002.From Σ it is standard to get the result of Szentmikl´ossy quoted ear-lier: since the compact space has no uncountable discrete subspace, it hascountable tightness. If it were not hereditarily Lindel¨of, it would have aright-separated subspace of size ℵ . But Σ implies it has an uncountablediscrete subspace, contradiction.Σ ′ is established by a minor variation of the forcing for Σ. A proofexists in the union of [L ] and [FTT]. Σ + , however, is not so clear, andhas not yet been proved from PFA( S )[ S ]. Thus, instead of using it to get ℵ -collectionwise Hausdorffness in locally compact normal spaces with no PAUL LARSON AND FRANKLIN D. TALL perfect pre-image of ω , as we did in an earlier version of this paper, weare instead quoting Lemma 3, which is a new result of the second author.Thus all we have to do is prove that PFA ++ (S)[S] implies Axiom R. Inorder to prove that PFA ++ (S)[S] implies Axiom R, we first note that astraightforward argument using the forcing Coll ( ω , X ) (whose conditionsare countable partial functions from ω to X , ordered by inclusion) showsthat PFA ++ (S) implies Axiom R ++ .It then suffices to prove: Lemma 6.
If Axiom R ++ holds and S is a Souslin tree, then Axiom R ++ still holds after forcing with S .Proof. First note that if X is a set, P is a c.c.c. forcing and τ is a P -namefor a tight cofinal subset of [ X ] <ω , then the set of a ∈ [ X ] <ω such thatevery condition in P forces that a is in the realization of τ is itself tightand cofinal. The tightness of this set is immediate. To see that it is cofinal,let b be any set in [ X ] <ω . Define sets b α ( α ≤ ω ) and σ α ( α < ω )recursively by letting σ α be a P -name for a member of the realization of τ containing b α and letting b α +1 be the set of members of X which are forcedby some condition in P to be in σ α . For limit ordinals α ≤ ω , let b α bethe union of the b β ( β < α ). Then b ω is forced by every condition in P tobe in τ .Since we are assuming that the Axiom of Choice holds, Axiom R ++ does not change if we require X to be an ordinal. Fix an ordinal γ and let ρ α ( α < ω ) be S -names for stationary subsets of [ γ ] <ω . Let T be a tightcofinal subset of [ γ ] <ω . For each countable ordinal α and each node s ∈ S ,let τ s,α be the set of countable subsets a of γ such that some condition in S extending s forces that a is in the realization of ρ α . Applying AxiomR ++ , we have a set Y ∈ [ γ ] <ω such that each P ( Y ) ∩ τ s,α is stationary in[ Y ] <ω .Since S is c.c.c., every club subset of [ Y ] <ω that exists after forcing with S includes a club subset of [ Y ] <ω existing in the ground model. Letting( ρ α ) G (for each α < ω ) be the realization of ρ α , we have by genericity thenthat after forcing with S , each P ( Y ) ∩ ( ρ α ) G will be stationary in [ Y ] <ω .This completes the proof of Theorem 1. AUL LARSON AND FRANKLIN D. TALL We do not know the answer to the following question; a positive answerwould likely enable us to dispense with Axiom R, and possibly with thesupercompact cardinal.
Problem.
Does MA ω imply every locally compact, hereditarily stronglycollectionwise Hausdorff space which does not include a perfect pre-imageof ω is paracompact?We also do not know whether in our main result, we can replace “perfectpre-image of ω ” by “copy of ω ”. Remark.
That PFA( S )[ S ] does not imply Axiom R is proved in [T ].The problem of finding in models of PFA( S )[ S ] necessary and sufficientconditions for locally compact normal spaces to be paracompact is studiedin [T ] by extending the methods of [B ] and this note. References [B ] Z. Balogh, Locally nice spaces under Martin’s axiom , Comment. Math. Univ.Carolin. (1983), 63–87.[B ] Z. Balogh, Locally nice spaces and Axiom R , Top. Appl. (2002), 335–341.[Ba] J.E. Baumgartner,
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PFA ( S )[ S ] implies there are no com-pact S -spaces (and more) , preprint.[G] G. Gruenhage, Some results on spaces having an orthobase or a base of subin-finite rank , Top. Proc. (1977), 151–159.[H] R. Hodel, Cardinal functions I , Handbook of Set-theoretic Topology (K. Kunenand J.E. Vaughan, eds.), North-Holland, Amsterdam, 1984.[L ] P. Larson, An S max variation for one Souslin tree , J. Symbolic Logic (1999),81–98.[L ] P. Larson, Notes on Todorcevic’s Erice lectures on forcing with a coherentSouslin tree , preprint. PAUL LARSON AND FRANKLIN D. TALL [LT] P. Larson and F.D. Tall, Locally compact perfectly normal spaces may all beparacompact , Fund. Math., 210 (2010), 285–300.[LTo] P. Larson and S. Todorˇcevi´c,
Katˇetov’s problem , Trans. Amer. Math. Soc (2002), 1783–1791.[Mi] T. Miyamoto, ω -Souslin trees under countable support iterations , Fund. Math. (1993), 257-261.[S] Z. Szentmikl´ossy, S -spaces and L -spaces under Martin’s Axiom , Topology (A.Cs´asz´ar, ed.), vol. II, North-Holland, 1980, pp. 1139–1146.[T ] F. D. Tall, Set-theoretic consistency results and topological theorems concern-ing the normal Moore space conjecture and related problems. Doctoral Disserta-tion, University of Wisconsin (Madison), 1969 , Dissertationes Math. (RozprawyMat.), (148) 1977.[T ] F. D. Tall, PFA(S)[S]: more mutually consistent topological consequences ofPFA and V=L , Canad. J. Math., to appear.[T ] F. D. Tall, PFA ( S )[ S ] and the Arhangel’ski˘ı-Tall problem , Top. Proc., to ap-pear.[T ] F. D. Tall, PFA ( S )[ S ] and locally compact normal spaces , submitted. Paul Larson, Department of Mathematics, Miami University, Oxford,Ohio 45056. e-mail address: [email protected] D. Tall, Department of Mathematics, University of Toronto,Toronto, Ontario M5S 2E4, CANADA e-mail address:e-mail address: