On the Hereditary Paracompactness of Locally Compact, Hereditarily Normal Spaces

Larson, Paul B.; Tall, Franklin D.

doi:10.4153/cmb-2014-010-3

Cited by 9 publications

(16 citation statements)

References 10 publications

(9 reference statements)

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“…In order to obtain a model of PFA(S)[S] in which Theorem 2.4 holds, we need to improve the model of [27] so as to not only have Axiom R but also:…”

Section: Strengthenings Of Pfa(s)[s]mentioning

confidence: 99%

“…This actually proves that {α ∈ ω 1 : N α ∩ κ ∈ S} is a stationary subset of ω 1 , because we could have put any cub of ω 1 as an element of M. Now assume Proof. This is an improvement over [27], which required a stronger axiom, Axiom R ++ , holding in the model. We will use t.u.b.…”

Section: Strengthenings Of Pfa(s)[s]mentioning

confidence: 99%

“…With the conclusion of [39] restored, [38], [27], and [40] are re-instated. We shall then proceed to improve the results of the two latter ones.…”

Section: Introductionmentioning

confidence: 99%

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Normality Versus Paracompactness in Locally Compact Spaces

Dow¹,

Tall²

2018

Can. j. math.

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This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the nonstationary ideal on ω 1 , as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of ω 1 .

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“…In order to obtain a model of PFA(S)[S] in which Theorem 2.4 holds, we need to improve the model of [27] so as to not only have Axiom R but also:…”

Section: Strengthenings Of Pfa(s)[s]mentioning

confidence: 99%

Section: Strengthenings Of Pfa(s)[s]mentioning

confidence: 99%

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Normality Versus Paracompactness in Locally Compact Spaces

Dow¹,

Tall²

2018

Can. j. math.

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“…In [14], [15], and [24], assuming the existence of a supercompact cardinal, a model of set theory is constructed, which we shall refer to as a model of PFA(S) [S]. We refer the reader to those papers for a discussion of what PFA(S)[S] is.…”

mentioning

confidence: 99%

“…In these papers various propositions concerning locally compact normal spaces are established in this model. We shall use: Lemma 3 [15]. In this model, locally compact hereditarily normal spaces which do not include a perfect pre-image of ω 1 are paracompact.…”

mentioning

confidence: 99%

Some observations on the Baireness of C(X) for a locally compact space X

Tall

2016

Topology and its Applications

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We prove some consistency results concerning the Moving Off Property for locally compact spaces, and thus the question of whether their function spaces are Baire. IntroductionThe Moving Off Property was introduced in [11] to characterize when C k (X) satisfies the Baire Category Theorem, for q-spaces X. Here we shall only be concerned with locally compact spaces (which are q), and so won't define q. We shall assume all spaces are Hausdorff. Definition.A moving off collection for a space X is a collection K of nonempty compact sets such that for each compact L, there is a K ∈ K disjoint from L. A space satisfies the Moving Off Property (MOP) if each moving off collection includes an infinite subcollection with a discrete open expansion.

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PFA(S)[S]and locally compact normal spaces

Tall

2014

Topology and its Applications

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We examine locally compact normal spaces in models of form PFA(S) [S], in particular characterizing paracompact, countably tight ones as those which include no perfect pre-image of ω 1 and in which all separable closed subspaces are Lindelöf.say PFA(S)[S] implies ϕ if ϕ holds whenever we force with S over a model of PFA(S), for S a coherent Souslin tree. We shall say ϕ holds in a model of form 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 on locally compact normal spaces because they entail many useful consequences of both PFA and V = L. We amalgamate here three previous preprints [45], [47], and [46] dealing with characterizing paracompactness and killing Dowker spaces in locally compact normal spaces, as well as with homogeneity in compact hereditarily normal spaces. Our proofs will avoid the difficult set-theoretic arguments in other papers on PFA(S)[S] by just quoting the familiar principles derived there, and so should be accessible to any set-theoretic topologist.The consequences of PFA(S)[S] we shall use and the references in which they are proved are: (Balogh's) (defined below) [19]; ℵ 1 -CWH (locally compact normal spaces are ℵ 1 -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 Lindelöf space is hereditarily separable [32]); FCℵ 1 -CWH (every first countable normal space is ℵ 1 -collectionwise Hausdorff [31]); OCA (Open Colouring Axiom [17]); b = ℵ 2 ([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]);

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On the Hereditary Paracompactness of Locally Compact, Hereditarily Normal Spaces

Abstract: Abstract. We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of ω 1 is hereditarily paracompact.

Cited by 9 publications

References 10 publications

Normality Versus Paracompactness in Locally Compact Spaces

Normality Versus Paracompactness in Locally Compact Spaces

Some observations on the Baireness of C(X) for a locally compact space X

PFA(S)[S]and locally compact normal spaces

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