Tightness of compact spaces is preserved by the $t$-equivalence relation

Oleg Okunev

A note on linear mappings between function spaces

Jan Baars (1993)

Commentationes Mathematicae Universitatis Carolinae

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Arhangel’skiǐ proved that if $X$ and $Y$ are completely regular spaces such that $C_{p} (X)$ and $C_{p} (Y)$ are linearly homeomorphic, then $X$ is pseudocompact if and only if $Y$ is pseudocompact. In addition he proved the same result for compactness, $σ$ -compactness and realcompactness. In this paper we prove that if $φ : C_{p} (X) \to C_{p} (X)$ is a continuous linear surjection, then $Y$ is pseudocompact provided $X$ is and if $φ$ is a continuous linear injection, then $X$ is pseudocompact provided $Y$ is. We also give examples that both statements...

Weak-bases and $D$ -spaces

Dennis K. Burke (2007)

Commentationes Mathematicae Universitatis Carolinae

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It is shown that certain weak-base structures on a topological space give a $D$ -space. This solves the question by A.V. Arhangel’skii of when quotient images of metric spaces are $D$ -spaces. A related result about symmetrizable spaces also answers a question of Arhangel’skii. Hence, quotient mappings, with compact fibers, from metric spaces have a $D$ -space image. What about quotient $s$ -mappings? Arhangel’skii and Buzyakova have shown that spaces with a point-countable base...

Displaying similar documents to “Tightness of compact spaces is preserved by the t -equivalence relation”

A note on linear mappings between function spaces

Weak-bases and D -spaces

Displaying similar documents to “Tightness of compact spaces is preserved by the $t$ -equivalence relation”

Weak-bases and $D$ -spaces