Locally Compact Normal Spaces in the Constructible Universe

Watson, W. Stephen

doi:10.4153/cjm-1982-078-8

Cited by 38 publications

(26 citation statements)

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“…In Cohen's original model for the negation of the Continuum Hypothesis (CH), and indeed in any model obtained by adding ℵ 2 Cohen reals to a model of ZFC, every locally compact normal space is ℵ 1 -cwH. This well-known fact can be shown by applying the proof technique of [W,Theorem 3] to [DTW,Theorem 1.4].…”

Section: Definition Let κ Be a Cardinal Number A Space X Is Weakly mentioning

confidence: 79%

Cardinal restrictions on some homogeneous compacta

Juhász

Nyikos

Szentmiklóssy

2005

Proc. Amer. Math. Soc.

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Abstract. We give restrictions on the cardinality of compact Hausdorff homogeneous spaces that do not use other cardinal invariants, but rather covering and separation properties. In particular, we show that it is consistent that every hereditarily normal homogeneous compactum is of cardinality c. We introduce property wD(κ), intermediate between the properties of being weakly κ-collectionwise Hausdorff and strongly κ-collectionwise Hausdorff, and show that if X is a compact Hausdorff homogeneous space in which every subspace has property wD(ℵ 1 ), then X is countably tight and hence of cardinality ≤ 2 c . As a corollary, it is consistent that such a space X is first countable and hence of cardinality c. A number of related results are shown and open problems presented.

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Section: Definition Let κ Be a Cardinal Number A Space X Is Weakly mentioning

confidence: 79%

Cardinal restrictions on some homogeneous compacta

Juhász

Nyikos

Szentmiklóssy

2005

Proc. Amer. Math. Soc.

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“…[Wat82] showed that V = L yielded an affirmative answer. By a difficult forcing argument, a consistent counterexample was constructed by Gruenhage and Koszmider [GK96a].…”

Section: Normality Versus Collectionwise Hausdorffnessmentioning

confidence: 99%

Some problems and techniques in set-theoretic topology

Tall¹

2011

Set Theory and Its Applications

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“…In [15], we used Lemma 1 of this section to show that, under V = L, locally compact normal spaces are collectionwise Hausdorff and that, under V -L, locally compact, metacompact normal spaces are paracompact.…”

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confidence: 99%

Separation in Countably Paracompact Spaces

Watson¹

1985

Transactions of the American Mathematical Society

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ABSTRACT. We study the question "Are discrete families of points separated in countably paracompact spaces?" in the class of first countable spaces and the class of separable spaces.Two of the main directions of research in general topology in the last thirty years have been the work of Jones, Bing, Tall, Fleissner, Nyikos and others motivated by the normal Moore space problem (when are discrete families separated in normal spaces?) and the work of Rudin, Zenor and others motivated by the Dowker space problem (what is the relation between normal and countably paracompact?). This paper considers the following question:(A) Are discrete families of points separated in countably paracompact spaces? There is a related question: (B) Are discrete families of points separated in normal spaces? This question has been studied in the class of first countable spaces and in the class of separable spaces (and so that is also where we consider question (A)) and has been more or less answered in those classes:In the class of first countable spaces, question (B) is independent of the axioms of set theory: V = L and other axioms imply yes (Fleissner, Tall [1,11]) and MA+-1CH implies no (Silver, Rothberger, Bing, Tall [11]). In the class of separable spaces, (B) is equivalent to the cardinal arithmetic 2^° < 2Hl (Jones, Heath [7,5]).The negative results consist of the construction of counterexamples which are countably paracompact (so that these negative results also apply to (A)) while the positive results consist of proofs for which normality appears essential. Fleissner [3] and Tall have asked whether it is consistent that, in the class of first countable spaces, the answer is yes to (A). Fleissner, Przymusinski and Reed [3,8,9] have asked whether it is possible to show that, in the class of separable spaces, (A) is equivalent to the cardinal arithmetic 2N° < 2Kl (Fleissner was able to extend Jones' short proof to show that, in the class of separable spaces, 2N° = Ni implies yes to (A)). We show that, in the class of first countable spaces, (A) is independent of the axioms of set theory (V = L implies yes) and that, in the class of separable spaces, (A) is equivalent to a set-theoretic statement whose equivalence with 2^° < 2Nl is a special case of a well-known open problem in set theory (Steprans, Jech and Prikry [6,10] have shown, independently, for example, that, if 2**° is a regular cardinal and there is no measurable cardinal in an inner model, then the equivalence holds).In this paper, a space is a regular topological space; a family {Aa:a < k} of subsets of a space is separated if there is a disjoint family {Oa: a < /c} of open sets

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Locally Compact Normal Spaces in the Constructible Universe

Cited by 38 publications

References 5 publications

Cardinal restrictions on some homogeneous compacta

Cardinal restrictions on some homogeneous compacta

Some problems and techniques in set-theoretic topology

Separation in Countably Paracompact Spaces

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