On subcompactness and countable  subcompactness of metrizable spaces in ZF

Kyriakos Keremedis

Displaying similar documents to “On subcompactness and countable subcompactness of metrizable spaces in ZF”

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.

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.

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 $𝔠$ .

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.

About $w c s$ -covers and $w c s^{*}$ -networks on the Vietoris hyperspace $ℱ (X)$

Luong Quoc Tuyen, Ong V. Tuyen, Phan D. Tuan, Nguzen X. Truc (2023)

Commentationes Mathematicae Universitatis Carolinae

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We study some generalized metric properties on the hyperspace $ℱ (X)$ of finite subsets of a space $X$ endowed with the Vietoris topology. We prove that $X$ has a point-star network consisting of (countable) $w c s$ -covers if and only if so does $ℱ (X)$ . Moreover, $X$ has a sequence of $w c s$ -covers with property $(P)$ which is a point-star network if and only if so does $ℱ (X)$ , where $(P)$ is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable....

Functionally countable subalgebras and some properties of the Banaschewski compactification

A. R. Olfati (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a zero-dimensional space and $C_{c} (X)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_{c}^{K} (X)$ (resp., $C_{c}^{ψ} (X))$ the set of all functions in $C_{c} (X)$ with compact (resp., pseudocompact) support. First, we observe that $C_{c}^{K} (X) = O_{c}^{β_{0} X ∖ X}$ (resp., $C_{c}^{ψ} (X) = M_{c}^{β_{0} X ∖ υ_{0} X}$ ), where $β_{0} X$ is the Banaschewski compactification of $X$ and $υ_{0} X$ is the $ℕ$ -compactification of $X$ . This implies that for an $ℕ$ -compact space $X$ , the intersection of all free maximal ideals in $C_{c} (X)$ is equal to $C_{c}^{K} (X)$ , i.e., $M_{c}^{β_{0} X ∖ X} = C_{c}^{K} (X)$ . By applying...

On non-normality points, Tychonoff products and Suslin number

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let a space $X$ be Tychonoff product $\prod_{α < τ} X_{α}$ of $τ$ -many Tychonoff nonsingle point spaces $X_{α}$ . Let Suslin number of $X$ be strictly less than the cofinality of $τ$ . Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $β X$ . In particular, this is true if $X$ is either $R^{τ}$ or $ω^{τ}$ and a cardinal $τ$ is infinite and not countably cofinal.

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.

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 the solvability of systems of linear equations over the ring $ℤ$ of integers

Horst Herrlich, Eleftherios Tachtsis (2017)

Commentationes Mathematicae Universitatis Carolinae

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We investigate the question whether a system ${(E_{i})}_{i \in I}$ of homogeneous linear equations over $ℤ$ is non-trivially solvable in $ℤ$ provided that each subsystem ${(E_{j})}_{j \in J}$ with $| J | \leq c$ is non-trivially solvable in $ℤ$ where $c$ is a fixed cardinal number such that $c < | I |$ . Among other results, we establish the following. (a) The answer is ‘No’ in the finite case (i.e., $I$ being finite). (b) The answer is ‘No’ in the denumerable case (i.e., $| I | = ℵ_{0}$ and $c$ a natural number). (c) The answer in case that $I$ is uncountable and $c \leq ℵ_{0}$ is ‘No...

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.

On butterfly-points in $β X$ , Tychonoff products and weak Lindelöf numbers

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be the Tychonoff product $\prod_{α < τ} X_{α}$ of $τ$ -many Tychonoff non-single point spaces $X_{α}$ . Let $p \in X^{*}$ be a point in the closure of some $G \subset X$ whose weak Lindelöf number is strictly less than the cofinality of $τ$ . Then we show that $β X ∖ {p}$ is not normal. Under some additional assumptions, $p$ is a butterfly-point in $β X$ . In particular, this is true if either $X = ω^{τ}$ or $X = R^{τ}$ and $τ$ is infinite and not countably cofinal.