To show that if every open set of a topological space X is paracompact, then every set in X is paracompact.

Question

I want to show that if every open set of a topological space $X$ is paracompact, then every set in $X$ is paracompact.

My idea was to first take an arbitrary set $A \subseteq X$ and an open conver $\{U_{\alpha}\}$, so this will also be an open cover of its interior, $A^{\circ}$, which will admit a locally finite open refinement, say $\{V_{\beta}\}$. I was wondering if it was possible to extend this to be a locally finite open refinement for $A$. Can someone point me in the right direction?

Answer 1

Unfortunately, the interior of $A$ is not so helpful--for instance, it could easily be empty. But there's something much simpler you can do: each $U_\alpha$ is of the form $V_\alpha\cap A$ for some $V_\alpha$ that is open in $X$. Now consider the open set $V=\bigcup V_\alpha$.

Details on how to finish are hidden below.

Since $V$ is paracompact by hypothesis, there is a locally finite open refinement of $(V_\alpha)$. Intersecting the sets in this refinement with $A$, we get a locally finite open refinement of $(U_\alpha)$.