On Szymański theorem on hereditary normality of $\beta \omega $

Sergei Logunov

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Displaying similar documents to “On Szymański theorem on hereditary normality of $β ω$ ”

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.

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.

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.

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...

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.

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 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.

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...

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.

Displaying similar documents to “On Szymański theorem on hereditary normality of β ω ”

Displaying similar documents to “On Szymański theorem on hereditary normality of $β ω$ ”