More reflections on compactness

Lúcia R. Junqueira; Franklin D. Tall

Displaying similar documents to “More reflections on compactness”

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 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.

A countably cellular topological group all of whose countable subsets are closed need not be $ℝ$ -factorizable

Mihail G. Tkachenko (2023)

Commentationes Mathematicae Universitatis Carolinae

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We construct a Hausdorff topological group $G$ such that $ℵ_{1}$ is a precalibre of $G$ (hence, $G$ has countable cellularity), all countable subsets of $G$ are closed and $C$ -embedded in $G$ , but $G$ is not $ℝ$ -factorizable. This solves Problem 8.6.3 from the book “Topological Groups and Related Structures" (2008) in the negative.

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...

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 $f : X^{2} \to [0, 1]$ with $Δ_{X} = f^{- 1} (0)$ , where $Δ_{X} = {(x, x) : x \in X}$ . In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most $𝔠$ .

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 $𝔠$ .

Locally functionally countable subalgebra of $ℛ (L)$

M. Elyasi, A. A. Estaji, M. Robat Sarpoushi (2020)

Archivum Mathematicum

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Let $L_{c} (X) = {f \in C (X) : \bar{C_{f}} = X}$ , where $C_{f}$ is the union of all open subsets $U \subseteq X$ such that $| f (U) | \leq ℵ_{0}$ . In this paper, we present a pointfree topology version of $L_{c} (X)$ , named $ℛ_{ℓ c} (L)$ . We observe that $ℛ_{ℓ c} (L)$ enjoys most of the important properties shared by $ℛ (L)$ and $ℛ_{c} (L)$ , where $ℛ_{c} (L)$ is the pointfree version of all continuous functions of $C (X)$ with countable image. The interrelation between $ℛ (L)$ , $ℛ_{ℓ c} (L)$ , and $ℛ_{c} (L)$ is examined. We show that $L_{c} (X) ≅ ℛ_{ℓ c} (𝔒 (X))$ for any space $X$ . Frames $L$ for which $ℛ_{ℓ c} (L) = ℛ (L)$ are characterized.

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.

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.

On hereditary normality of $ω^{*}$ , Kunen points and character $ω_{1}$

Sergei Logunov (2021)

Commentationes Mathematicae Universitatis Carolinae

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We show that $ω^{*} ∖ {p}$ is not normal, if $p$ is a limit point of some countable subset of $ω^{*}$ , consisting of points of character $ω_{1}$ . Moreover, such a point $p$ is a Kunen point and a super Kunen point.