Locally nice spaces and Axiom R

Balogh, Zoltán

doi:10.1016/s0166-8641(01)00286-3

Cited by 22 publications

(38 citation statements)

References 3 publications

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“…The compatibility of "locally countable subspaces of size ℵ 1 in a compact countably tight space are σ-discrete" with "normal first countable spaces are collectionwise Hausdorff" enables us to strengthen a variety of results of Balogh [1] and other authors, in particular proving Theorem 1, which we shall soon establish.…”

Section: Topological Applicationsmentioning

confidence: 76%

“…We drop two of these conditions and get: Theorem 1 follows easily. Theorem 17 answers a question Balogh asked for manifolds in [1]. Theorem 17 will follow immediately from the following lemma, which is essentially due to Balogh [1] (see the proof of his Theorem 3.3).…”

Section: Topological Applicationsmentioning

confidence: 90%

“…In Section 2, we shall prove Theorem 1 and other topological consequences of Theorems 2 and 3. Section 3 provides an alternative, more direct, path to Theorem 1 for the reader -perhaps a set theorist-who is not interested in more general results such as Theorem 3.IV.6 and doesn't want to wade through or depend on [1]. In Section 4, we shall apply Theorem 2 to Whitehead groups.…”

Section: Theorem 7 [39] Pfa(s)[s] Implies Every Compact Countably Timentioning

confidence: 99%

“…Balogh [1] proved under MA + ∼CH that connected, locally compact, locally hereditarily Lindelöf, hereditarily normal collectionwise Hausdorff spaces are paracompact if and only if they do not include a perfect pre-image of ω 1 . We drop two of these conditions and get: Theorem 1 follows easily.…”

Section: Topological Applicationsmentioning

confidence: 99%

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Locally compact perfectly normal spaces may all be paracompact

Larson

Tall

2010

Fund. Math.

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Abstract. We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space with a hereditarily normal square is metrizable. We also solve a problem raised by the second author, proving it consistent with ZFC that every first countable hereditarily normal countable chain condition space is hereditarily separable.

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Section: Topological Applicationsmentioning

confidence: 76%

Section: Topological Applicationsmentioning

confidence: 90%

Section: Theorem 7 [39] Pfa(s)[s] Implies Every Compact Countably Timentioning

confidence: 99%

Section: Topological Applicationsmentioning

confidence: 99%

See 2 more Smart Citations

Locally compact perfectly normal spaces may all be paracompact

Larson

Tall

2010

Fund. Math.

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“…So suppose that a compact K is not hereditarily Lindelöf and so has a right separated uncountable sequence. Such a sequence is locally countable and so by [3] if K is countably tight, this sequence is a countable union of discrete subspaces. In any case K has an uncountable discrete space, so (1) can be applied.…”

Section: Totally Disconnected Nonreflection In All Continuous Imagesmentioning

confidence: 99%

On the problem of compact totally disconnected reflection of nonmetrizability

Koszmider

2016

Topology and its Applications

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Abstract. We construct a ZFC example of a nonmetrizable compact space K such that every totally disconnected closed subspace L ⊆ K is metrizable. In fact, the construction can be arranged so that every nonmetrizable compact subspace may be of fixed big dimension. Then we focus on the problem if a nonmetrizable compact space K must have a closed subspace with a nonmetrizable totally disconnected continuous image. This question has several links with the the structure of the Banach space C(K), for example, by Holsztyński's theorem, if K is a counterexample, then C(K) contains no isometric copy of a nonseparable Banach space C(L) for L totally disconnected. We show that in the literature there are diverse consistent counterexamples, most eliminated by Martin's axiom and the negation of the continuum hypothesis, but some consistent with it. We analyze the above problem for a particular class of spaces. OCA+MA however, implies the nonexistence of any counterexample in this class but the existence of some other absolute example remains open.

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The list-chromatic number and the coloring number of uncountable graphs

Usuba

2023

Isr. J. Math.

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Locally nice spaces and Axiom R

Cited by 22 publications

References 3 publications

Locally compact perfectly normal spaces may all be paracompact

Locally compact perfectly normal spaces may all be paracompact

On the problem of compact totally disconnected reflection of nonmetrizability

The list-chromatic number and the coloring number of uncountable graphs

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