Displaying similar documents to “On nowhere first-countable compact spaces with countable π -weight”

On Szymański theorem on hereditary normality of β ω

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

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We discuss the following result of A. Szymański in “Retracts and non-normality points" (2012), Corollary 3.5.: If F is a closed subspace of ω * and the π -weight of F is countable, then every nonisolated point of F is a non-normality point of ω * . We obtain stronger results for all types of points, excluding the limits of countable discrete sets considered in “Some non-normal subspaces of the Čech–Stone compactification of a discrete space” (1980) by A. Błaszczyk and A. Szymański. Perhaps...

A note on star Lindelöf, first countable and normal spaces

Wei-Feng Xuan (2017)

Mathematica Bohemica

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A topological space X is said to be star Lindelöf if for any open cover 𝒰 of X there is a Lindelöf subspace A ⊂ X such that St ( A , 𝒰 ) = X . The “extent” e ( X ) of X is the supremum of the cardinalities of closed discrete subsets of X . We prove that under V = L every star Lindelöf, first countable and normal space must have countable extent. We also obtain an example under MA + ¬ CH , which shows that a star Lindelöf, first countable and normal space may not have countable extent.

An observation on spaces with a zeroset diagonal

Wei-Feng Xuan (2020)

Mathematica Bohemica

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We say that a space X has the discrete countable chain condition (DCCC for short) if every discrete family of nonempty open subsets of X is countable. A space X has a zeroset diagonal if there is a continuous mapping f : X 2 → [ 0 , 1 ] with Δ X = f - 1 ( 0 ) , where Δ X = { ( x , x ) : x ∈ X } . In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most 𝔠 .

More reflections on compactness

Lúcia R. Junqueira, Franklin D. Tall (2003)

Fundamenta Mathematicae

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We consider the question of when X M = X , where X M is the elementary submodel topology on X ∩ M, especially in the case when X M is compact.

A nice subclass of functionally countable spaces

Vladimir Vladimirovich Tkachuk (2018)

Commentationes Mathematicae Universitatis Carolinae

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A space X is functionally countable if f ( X ) is countable for any continuous function f : X → ℝ . We will call a space X exponentially separable if for any countable family ℱ of closed subsets of X , there exists a countable set A ⊂ X such that A ∩ ⋂ 𝒢 ≠ ∅ whenever 𝒢 ⊂ ℱ and ⋂ 𝒢 ≠ ∅ . Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable...

On a question of C c ( X )

A. R. Olfati (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this short article we answer the question posed in Ghadermazi M., Karamzadeh O.A.S., Namdari M., On the functionally countable subalgebra of C ( X ) , Rend. Sem. Mat. Univ. Padova 129 (2013), 47–69. It is shown that C c ( X ) is isomorphic to some ring of continuous functions if and only if υ 0 X is functionally countable. For a strongly zero-dimensional space X , this is equivalent to say that X is functionally countable. Hence for every P -space it is equivalent to pseudo- ℵ 0 -compactness.

A note on spaces with countable extent

Yan-Kui Song (2017)

Commentationes Mathematicae Universitatis Carolinae

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Let P be a topological property. A space X is said to be star P if whenever 𝒰 is an open cover of X , there exists a subspace A ⊆ X with property P such that X = S t ( A , 𝒰 ) . In this note, we construct a Tychonoff pseudocompact SCE-space which is not star Lindelöf, which gives a negative answer to a question of Rojas-Sánchez and Tamariz-Mascarúa.

On n -thin dense sets in powers of topological spaces

Adam Bartoš (2016)

Commentationes Mathematicae Universitatis Carolinae

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A subset of a product of topological spaces is called n -thin if every its two distinct points differ in at least n coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable T 3 space X without isolated points such that X n contains an n -thin dense subset, but X n + 1 does not contain any n -thin dense subset. We also observe that part of the construction can be carried out under MA.

Spaces with property ( D C ( ω 1 ) )

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove that if X is a first countable space with property ( D C ( ω 1 ) ) and with a G δ -diagonal then the cardinality of X is at most 𝔠 . We also show that if X is a first countable, DCCC, normal space then the extent of X is at most 𝔠 .

On subcompactness and countable subcompactness of metrizable spaces in ZF

Kyriakos Keremedis (2022)

Commentationes Mathematicae Universitatis Carolinae

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We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space 𝐗 = ( X , T ) is countably compact if and only if it is countably subcompact relative to T . (iii) For every metrizable space 𝐗 = ( X , T ) , the following are equivalent: (a) 𝐗 is compact; (b) for every open filter ℱ of 𝐗 , ⋂ { F ¯ : F ∈ ℱ } ≠ ∅ ; (c) 𝐗 is subcompact relative to T . We also show: (iv) The negation of each of the statements, (a) every countably subcompact...

A new Lindelöf space with points G δ

Alan S. Dow (2015)

Commentationes Mathematicae Universitatis Carolinae

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We prove that ♦ * implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality 2 ℵ 1 which has points G δ . In addition, this space has the property that it need not be Lindelöf after countably closed forcing.