Diagonals and discrete subsets of squares

Dennis Burke; Vladimir Vladimirovich Tkachuk

Weak extent in normal spaces

Ronnie Levy, Mikhail Matveev (2005)

Commentationes Mathematicae Universitatis Carolinae

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If $X$ is a space, then the $we (X)$ of $X$ is the cardinal $min {α :$ If $𝒰$ is an open cover of $X$ , then there exists $A \subseteq X$ such that $| A | = α$ and $St (A, 𝒰) = X}$ . In this note, we show that if $X$ is a normal space such that $| X | = 𝔠$ and $we (X) = ω$ , then $X$ does not have a closed discrete subset of cardinality $𝔠$ . We show that this result cannot be strengthened in ZFC to get that the extent of $X$ is smaller than $𝔠$ , even if the condition that $we (X) = ω$ is replaced by the stronger condition that $X$ is separable.

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...

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Weak extent in normal spaces

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