$S$-Paracompactness and $S_2$-Paracompactness.

Abstract

A topological space $X$ is an $S$-{\it paracompact} if there exists a bijective function $f$ from $X$ onto a paracompact space $Y$ such that for every separable subspace $A$ of $X$ the restriction map $f|_A$ from $A$ onto $f(A)$ is a homeomorphism. Moreover, if $Y$ is Hausdorff, then $X$ is called $S_2$-{\it paracompact}.
We investigate these two properties.