When is $\mathbf {N}$ Lindelöf?

Horst Herrlich; George E. Strecker

Displaying similar documents to “When is $𝐍$ Lindelöf?”

On the Set-Theoretic Strength of Countable Compactness of the Tychonoff Product $2^{ℝ}$

Eleftherios Tachtsis (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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We work in ZF set theory (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) and show the following: 1. The Axiom of Choice for well-ordered families of non-empty sets ( $A C^{W O}$ ) does not imply “the Tychonoff product $2^{ℝ}$ , where 2 is the discrete space 0,1, is countably compact” in ZF. This answers in the negative the following question from Keremedis, Felouzis, and Tachtsis [Bull. Polish Acad. Sci. Math. 55 (2007)]: Does the Countable Axiom of Choice for families of non-empty sets...

Three small results on normal first countable linearly H-closed spaces

Mathieu Baillif (2022)

Commentationes Mathematicae Universitatis Carolinae

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We use topological consequences of , and proved by other authors to show that normal first countable linearly H-closed spaces with various additional properties are compact in these models.

On the Compactness and Countable Compactness of $2^{ℝ}$ in ZF

Kyriakos Keremedis, Evangelos Felouzis, Eleftherios Tachtsis (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements " $2^{ℝ}$ is countably compact" and " $2^{ℝ}$ is compact"

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...

Counting models of set theory

Ali Enayat (2002)

Fundamenta Mathematicae

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Let T denote a completion of ZF. We are interested in the number μ(T) of isomorphism types of countable well-founded models of T. Given any countable order type τ, we are also interested in the number μ(T,τ) of isomorphism types of countable models of T whose ordinals have order type τ. We prove: (1) Suppose ZFC has an uncountable well-founded model and $κ \in ω \cup ℵ ₀, ℵ ₁, 2^{ℵ ₀}$ . There is some completion T of ZF such that μ(T) = κ. (2) If α <ω₁ and μ(T,α) > ℵ₀, then $μ (T, α) = 2^{ℵ ₀}$ . (3) If α < ω₁ and T ⊢ V ≠ OD,...

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 solution to Comfort's question on the countable compactness of powers of a topological group

Artur Hideyuki Tomita (2005)

Fundamenta Mathematicae

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In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $α \leq 2^{}$ , a topological group G such that $G^{γ}$ is countably compact for all cardinals γ < α, but $G^{α}$ is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M A_{c o u n t a b l e}$ . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M A_{c o u n t a b l e}$ . However, the question has...

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.

$p$ -sequential like properties in function spaces

Salvador García-Ferreira, Angel Tamariz-Mascarúa (1994)

Commentationes Mathematicae Universitatis Carolinae

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We introduce the properties of a space to be strictly $WFU (M)$ or strictly $SFU (M)$ , where $\emptyset \neq M \subset ω^{*}$ , and we analyze them and other generalizations of $p$ -sequentiality ( $p \in ω^{*}$ ) in Function Spaces, such as Kombarov’s weakly and strongly $M$ -sequentiality, and Kocinac’s $WFU (M)$ and $SFU (M)$ -properties. We characterize these in $C_{π} (X)$ in terms of cover-properties in $X$ ; and we prove that weak $M$ -sequentiality is equivalent to $WFU (L (M))$ -property, where $L (M) = {^{λ} p : λ < ω_{1}$ and $p \in M}$ , in the class of spaces which are $p$ -compact for every $p \in M \subset ω^{*}$ ; and that $C_{π} (X)$ is a $WFU (L (M))$ -space iff...

Combinatorics of open covers (VII): Groupability

Ljubiša D. R. Kočinac, Marion Scheepers (2003)

Fundamenta Mathematicae

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We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a $T_{31 / 2}$ -space. In [9] we showed that $C_{p} (X)$ has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. $C_{p} (X)$ has countable fan tightness and the Reznichenko...

Some versions of second countability of metric spaces in ZF and their role to compactness

Kyriakos Keremedis (2018)

Commentationes Mathematicae Universitatis Carolinae

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In the realm of metric spaces we show in ZF that: (i) A metric space is compact if and only if it is countably compact and for every $ε > 0$ , every cover by open balls of radius $ε$ has a countable subcover. (ii) Every second countable metric space has a countable base consisting of open balls if and only if the axiom of countable choice restricted to subsets of $ℝ$ holds true. (iii) A countably compact metric space is separable if and only if it is second countable.

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.

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 star covering properties related to countable compactness and pseudocompactness

Marcelo D. Passos, Heides L. Santana, Samuel G. da Silva (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove a number of results on star covering properties which may be regarded as either generalizations or specializations of topological properties related to the ones mentioned in the title of the paper. For instance, we give a new, entirely combinatorial proof of the fact that $Ψ$ -spaces constructed from infinite almost disjoint families are not star-compact. By going a little further we conclude that if $X$ is a star-compact space within a certain class, then $X$ is neither first countable...

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 $𝔠$ .

Displaying similar documents to “When is 𝐍 Lindelöf?”

Displaying similar documents to “When is $𝐍$ Lindelöf?”