Normal Moore Spaces in the Constructible Universe

Fleissner, William G.

doi:10.2307/2039914

Cited by 35 publications

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“…Then the sets A, of Lemma 2.4 are all stationary. It is easily seen that for all/, g: k -"to and a G k, we have/1^ D (a + 1) = Ag n (a + 1) whenever f\ a = g | a. Consequently, the sets Af, for /: k -"to, form what Fleissner calls a stationary system, and we can apply Lemma 2 of [2] to conclude that there exists S G k and <I>(a): a ->"u, for a G S, such that for each /: k ^"u, the set {a: <I>(a) = f\ a) is a stationary set contained in Af.…”

Section: (V -L) For a Regular K Tals(< K) =» Tals(k)mentioning

confidence: 93%

“…If we use Lemma 2.4 above to substitute for Lemma 1 of [2], the proof of Lemma 2.5 is almost identical to that of Lemma 3 of [2]. We assume that TALS(< k) holds but there is a totally analytic space X with underlying set k such that X is not /1-left-separated.…”

Section: (V -L) For a Regular K Tals(< K) =» Tals(k)mentioning

confidence: 99%

“…To obtain our results, we use Gödel's constructibility axiom V -L. Previous to the results below, G. M. Reed has shown that, under V = L, each first countable normal space with every subset an /""-set is /""-discrete [8], and R. W. Hansell has shown that, under V = L, a totally analytic space X of character < to, is a-discrete provided that the product of X with the irrationals is normal [5]. Like Reed and Hansell, we rely on W. G. Fleissner's techniques and results in [2] (dealing with the seemingly unrelated topic of collectionwise Hausdorffness of normal spaces of character =£ «, under V = L).…”

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confidence: 99%

“…The proof of Theorem 2.1 is a modification of Fleissner's proof of the main result in [2]. We shall give some lemmas from which the theorem follows; for a few of these lemmas, we do not give a proof but only indicate the changes that are needed in the proofs of the corresponding results in [2].…”

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confidence: 99%

“…In [2], mappings A -co, were used to code open separations of subsets of a discrete subset A of a space X (of character < u, ). Here we use mappings X -"u to code partitions of X into analytic sets.…”

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confidence: 99%

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Totally analytic spaces under 𝑉=𝐿

Balogh¹,

Junnila²

1983

Proc. Amer. Math. Soc.

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The following results obtain under the axiom of constructibility ( V -L): Assume that every subset of a topological space -V is analytic. Then A' is o-left-separatcd. Moreover, if the character of X is « u,, then X is o-discrete.Assume that X is a perfectly normal space of character « u, such that every subset of X belongs to the o-algebra generated by the analytic subsets of X. Then X is o-discrete.1. Introduction. By an analytic subset of a topological space A", we mean a set that can be obtained from a family of closed subsets of X by the A -operation, i.e., a set -f the form LI " D c FM", where the sets F,, s G^"(j, are closed in X. We say that X is a totally analytic space if each subset of X is analytic. This notion generalizes that of a space in which each subset is an F"-set.A space X is o-discrete (Fa-discrete) if X is the countable union of discrete (and closed) subspaces. X is left-separated if there exists a well-order < on X such that all the initial segments {y G X: y < x}, x G A", are closed in X. We say that X is o-left-separated if X is the countable union of left-separated subspaces. Note that the concept of an /^-left-separated space is superfluous, since such a space is actually left-separated; in particular, every /"-discrete space is left-separated.The character of a space X is the least cardinal number k such that every point of X has a neighbourhood-base consisting of at most k sets; the character of X is

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Section: (V -L) For a Regular K Tals(< K) =» Tals(k)mentioning

confidence: 93%

Section: (V -L) For a Regular K Tals(< K) =» Tals(k)mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Totally analytic spaces under 𝑉=𝐿

Balogh¹,

Junnila²

1983

Proc. Amer. Math. Soc.

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On finite-dimensional nonmetrizable manifolds

Fedorchuk¹

1995

J Math Sci

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Problems arising from Balogh's “Locally nice spaces under Martin's Axiom”

Tall

2005

Topology and its Applications

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We discuss recent advances extending the paper mentioned in the title, in particular, the consistency proof for "all locally compact perfectly normal spaces are paracompact". As well, we point out mistakes in the published literature concerning metrizability of hereditarily normal manifolds, and set out an approach toward proving the consistency of such metrizability for those of dimension greater than 1.  2004 Elsevier B.V. All rights reserved.MSC: primary 54D45, 54D20, 54D15, 03E35, 54A35; secondary 03E35, 03E50, 54E35, 57N99

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Normal Moore Spaces in the Constructible Universe

Abstract: ABSTRACT.Assuming the axiom of constructibility, points in closed discrete subspaces of certain normal spaces can be simultaneously separated. This is a partial result towards the normal Moore space conjecture.

Cited by 35 publications

References 0 publications

Totally analytic spaces under 𝑉=𝐿

Totally analytic spaces under 𝑉=𝐿

On finite-dimensional nonmetrizable manifolds

Problems arising from Balogh's “Locally nice spaces under Martin's Axiom”

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