Addition theorems for dense subspaces

Aleksander V. Arhangel'skii

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Displaying similar documents to “Addition theorems for dense subspaces”

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 $f : X \to ℝ$ . We will call a space $X$ exponentially separable if for any countable family $ℱ$ of closed subsets of $X$ , there exists a countable set $A \subset X$ such that $A \cap ⋂ 𝒢 \neq \emptyset$ whenever $𝒢 \subset ℱ$ and $⋂ 𝒢 \neq \emptyset$ . 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...

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 $A \subset X$ 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 $MA + \neg CH$ , which shows that a star Lindelöf, first countable and normal space may not have countable extent.

On universality of countable and weak products of sigma hereditarily disconnected spaces

Taras Banakh, Robert Cauty (2001)

Fundamenta Mathematicae

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Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power $X^{ω}$ of any subspace X ⊂ Y is not universal for the class ₂ of absolute $G_{δ σ}$ -sets; moreover, if Y is an absolute $F_{σ δ}$ -set, then $X^{ω}$ contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute $G_{δ}$ -set, then $X^{ω}$ contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable...

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 $𝐗$ , $⋂ {\bar{F} : F \in ℱ} \neq \emptyset$ ; (c) $𝐗$ is subcompact relative to $T$ . We also show: (iv) The negation of each of the statements, (a) every countably subcompact...

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 $A \subseteq X$ 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.

Non-normality points and nice spaces

Sergei Logunov (2021)

Commentationes Mathematicae Universitatis Carolinae

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J. Terasawa in " $β X - {p}$ are non-normal for non-discrete spaces $X$ " (2007) and the author in “On non-normality points and metrizable crowded spaces” (2007), independently showed for any metrizable crowded space $X$ that each point $p$ of its Čech–Stone remainder $X^{*}$ is a non-normality point of $β X$ . We introduce a new class of spaces, named nice spaces, which contains both of Sorgenfrey line and every metrizable crowded space. We obtain the result above for every nice space.

Exponential separability is preserved by some products

Vladimir Vladimirovich Tkachuk (2022)

Commentationes Mathematicae Universitatis Carolinae

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We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a $σ$ -compact crowded space in which all countable subspaces are scattered. If $X$ is a Lindelöf space and every $Y \subset X$ with $| Y | \leq 2^{ω_{1}}$ is scattered, then $X$ is functionally countable; if every $Y \subset X$ with $| Y | \leq 2^{𝔠}$ is scattered,...

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.

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...

Some results on semi-stratifiable spaces

Wei-Feng Xuan, Yan-Kui Song (2019)

Mathematica Bohemica

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We study relationships between separability with other properties in semi-stratifiable spaces. Especially, we prove the following statements: (1) If $X$ is a semi-stratifiable space, then $X$ is separable if and only if $X$ is $D C (ω_{1})$ ; (2) If $X$ is a star countable extent semi-stratifiable space and has a dense metrizable subspace, then $X$ is separable; (3) Let $X$ be a $ω$ -monolithic star countable extent semi-stratifiable space. If $t (X) = ω$ and $d (X) \leq ω_{1}$ , then $X$ is hereditarily separable. Finally, we prove that for...

Spaces with star countable extent

A. D. Rojas-Sánchez, Angel Tamariz-Mascarúa (2016)

Commentationes Mathematicae Universitatis Carolinae

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For a topological property $P$ , we say that a space $X$ is star $P$ if for every open cover $𝒰$ of the space $X$ there exists $A \subset X$ such that $s t (A, 𝒰) = X$ . We consider space with star countable extent establishing the relations between the star countable extent property and the properties star Lindelöf and feebly Lindelöf. We describe some classes of spaces in which the star countable extent property is equivalent to either the Lindelöf property or separability. An example is given of a Tychonoff star Lindelöf...

Characterizations of $z$ -Lindelöf spaces

Ahmad Al-Omari, Takashi Noiri (2017)

Archivum Mathematicum

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A topological space $(X, τ)$ is said to be $z$ -Lindelöf [1] if every cover of $X$ by cozero sets of $(X, τ)$ admits a countable subcover. In this paper, we obtain new characterizations and preservation theorems of $z$ -Lindelöf spaces.

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.