Advanced Search

A topological space $X$ is said to be $e$-separable if $X$ has a $\sigma$-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.

A space $X$ is said to be if there is a basis $ℬ$ for $X$ with $\left|ℬ\right|=w\left(X\right)$ 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 $ℱ$.