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Monotonically normal $e$ -separable spaces may not be perfect

John E. Porter — 2018

Commentationes Mathematicae Universitatis Carolinae

A topological space $X$ is said to be $e$ -separable if $X$ has a $σ$ -closed-discrete dense subset. Recently, G. Gruenhage and D. Lutzer showed that $e$ -separable PIGO spaces are perfect and asked if $e$ -separable monotonically normal spaces are perfect in general. The main purpose of this article is to provide examples of $e$ -separable monotonically normal spaces which are not perfect. Extremely normal $e$ -separable spaces are shown to be stratifiable.

Strongly base-paracompact spaces

John E. Porter — 2003

Commentationes Mathematicae Universitatis Carolinae

A space $X$ is said to be if there is a basis $ℬ$ for $X$ with $| ℬ | = w (X)$ such that every open cover of $X$ has a star-finite open refinement by members of $ℬ$ . Strongly paracompact spaces which are strongly base-paracompact are studied. Strongly base-paracompact spaces are shown have a family of functions $ℱ$ with cardinality equal to the weight such that every open cover has a locally finite partition of unity subordinated to it from $ℱ$ .

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Monotonically normal e -separable spaces may not be perfect

Strongly base-paracompact spaces

Monotonically normal $e$ -separable spaces may not be perfect