Let $ X $ be a topological space, paracompact, $T_2$. Let $S \subset X$, with $S$ is generalized-$F_{\sigma}$ then $ S $ is paracompact.

Question

Let $ X $ be a topological space, paracompact, $T_2$, and $S \subset X$ with $S$ is generalized-$F_{\sigma}$ then $ S $ is paracompact.

A subset S of a topological space $X$ is called a generalized-$F_{\sigma}$ set in $X$ if for all open $G \subset X$ with $S \subset G$, there exists an $F_{\sigma}$-set $F \subset X$ such that $S \subset F \subset G$.

We have to $F$ an $F_{\sigma}$-set in a paracompact space is paracompact, but how can I say that $ S $ is also paracompact, does it inherit the paracompact of $ F $?

Answer 1

Let $\mathscr{U}$ be a relatively open cover of $S$. For each $U\in\mathscr{U}$ let $V_U$ be an open set in $X$ such that $U=V_U\cap S$, and let $\mathscr{V}=\{V_U:U\in\mathscr{U}\}$. There is an $F_\sigma$-set $F$ such that $S\subseteq F\subseteq\bigcup\mathscr{V}$, and $F$ is paracompact, so $\{V_U\cap F:U\in\mathscr{U}\}$ has a locally finite $F$-open refinement $\mathscr{R}$ that covers $F$. Now just check that $\{R\cap S:R\in\mathscr{R}\}$ is a locally finite $S$-open refinement of $\mathscr{U}$ that covers $S$.