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

Mathieu Baillif

Displaying similar documents to “Three small results on normal first countable linearly H-closed spaces”

When is $𝐍$ Lindelöf?

Horst Herrlich, George E. Strecker (1997)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) $ℕ$ is a Lindelöf space, (2) $ℚ$ is a Lindelöf space, (3) $ℝ$ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of $ℝ$ is separable, (6) in $ℝ$ , a point $x$ is in the closure of a set $A$ iff there exists a sequence in $A$ that converges to $x$ , (7) a function $f : ℝ \to ℝ$ is continuous at a point $x$ iff $f$ is sequentially continuous...

Point-countable π-bases in first countable and similar spaces

V. V. Tkachuk (2005)

Fundamenta Mathematicae

Similarity:

It is a classical result of Shapirovsky that any compact space of countable tightness has a point-countable π-base. We look at general spaces with point-countable π-bases and prove, in particular, that, under the Continuum Hypothesis, any Lindelöf first countable space has a point-countable π-base. We also analyze when the function space $C_{p} (X)$ has a point-countable π -base, giving a criterion for this in terms of the topology of X when l*(X) = ω. Dealing with point-countable π-bases makes...

In search for Lindelöf $C_{p}$ ’s

Raushan Z. Buzyakova (2004)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

It is shown that if $X$ is a first-countable countably compact subspace of ordinals then $C_{p} (X)$ is Lindelöf. This result is used to construct an example of a countably compact space $X$ such that the extent of $C_{p} (X)$ is less than the Lindelöf number of $C_{p} (X)$ . This example answers negatively Reznichenko’s question whether Baturov’s theorem holds for countably compact spaces.

Spaces with star countable extent

A. D. Rojas-Sánchez, Angel Tamariz-Mascarúa (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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

Similarity:

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,...

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

Kyriakos Keremedis (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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.

On pseudocompactness and related notions in ZF

Kyriakos Keremedis (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We study in ZF and in the class of $T_{1}$ spaces the web of implications/ non-implications between the notions of pseudocompactness, light compactness, countable compactness and some of their ZFC equivalents.

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

Similarity:

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...

Combinatorics of open covers (VII): Groupability

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

Fundamenta Mathematicae

Similarity:

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...