Countable 1-transitive coloured linear orderings II

G. Campero-Arena; J. K. Truss

Displaying similar documents to “Countable 1-transitive coloured linear orderings II”

In search for Lindelöf $C_{p}$ ’s

Raushan Z. Buzyakova (2004)

Commentationes Mathematicae Universitatis Carolinae

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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

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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...

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

V. V. Tkachuk (2005)

Fundamenta Mathematicae

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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...

A note on transitively $D$ -spaces

Liang-Xue Peng (2011)

Czechoslovak Mathematical Journal

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In this note, we show that if for any transitive neighborhood assignment $φ$ for $X$ there is a point-countable refinement $ℱ$ such that for any non-closed subset $A$ of $X$ there is some $V \in ℱ$ such that $| V \cap A | \geq ω$ , then $X$ is transitively $D$ . As a corollary, if $X$ is a sequential space and has a point-countable $w c s^{*}$ -network then $X$ is transitively $D$ , and hence if $X$ is a Hausdorff $k$ -space and has a point-countable $k$ -network, then $X$ is transitively $D$ . We prove that if $X$ is a countably compact sequential space and...

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...

On the product of a compact space with an hereditarily absolutely countably compact space

Maddalena Bonanzinga (1997)

Commentationes Mathematicae Universitatis Carolinae

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We show that the product of a compact, sequential $T_{2}$ space with an hereditarily absolutely countably compact $T_{3}$ space is hereditarily absolutely countably compact, and further that the product of a compact $T_{2}$ space of countable tightness with an hereditarily absolutely countably compact $ω$ -bounded $T_{3}$ space is hereditarily absolutely countably compact.

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...

A first countable supercompact Hausdorff space with a closed $G_{δ}$ nonsupercompact subspace

Murray G. Bell (1980)

Colloquium Mathematicae

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When is $𝐍$ Lindelöf?

Horst Herrlich, George E. Strecker (1997)

Commentationes Mathematicae Universitatis Carolinae

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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...