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

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?

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 $$V=\bigcup V_\alpha$$.
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 $$(U_\alpha)$$.