A new Lindelöf space with points $G_\delta $

Alan S. Dow

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Displaying similar documents to “A new Lindelöf space with points $G_{δ}$ ”

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.

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 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.

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 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.

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.

$C^{*}$ -points vs $P$ -points and $P^{♭}$ -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space $X$ , the point $p \in X$ is called a $C^{*}$ -point if every real-valued continuous function on $C ∖ {p}$ can be extended continuously to $p$ . Every point in an extremally disconnected space is a $C^{*}$ -point. A classic example is the space $𝐖^{*} = ω_{1} + 1$ consisting of the countable ordinals together with $ω_{1}$ . The point $ω_{1}$ is known to be a $C^{*}$ -point as well as a $P$ -point. We supply a characterization of $C^{*}$ -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Continuous images of Lindelöf $p$ -groups, $σ$ -compact groups, and related results

Aleksander V. Arhangel'skii (2019)

Commentationes Mathematicae Universitatis Carolinae

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It is shown that there exists a $σ$ -compact topological group which cannot be represented as a continuous image of a Lindelöf $p$ -group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf $p$ -groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space $Y$ is a continuous image of a Lindelöf $p$ -group, then there exists a covering $γ$ of $Y$ by dyadic compacta such that $| γ | \leq 2^{ω}$ . We also show that if a homogeneous compact space $Y$ is...

Displaying similar documents to “A new Lindelöf space with points G δ ”

Displaying similar documents to “A new Lindelöf space with points $G_{δ}$ ”