# countable union of closed paracompact subspaces

This is an exercise of Munkres topology section 41.

Let $$X$$ be a regular space. If $$X$$ is a countable union of closed paracompact subspaces of $$X$$ whose interiors cover $$X$$, show $$X$$ is paracompact.

I Know this lemma:

Let $$X$$ be a regular space. Then TFAE (note: a refinement must itself also be a cover).

1. Every open cover of $$X$$ has a locally finite open refinement. (i.e. $$X$$ is paracompact)
2. Every open cover of $$X$$ has a $$\sigma$$-locally finite open refinement (where a family of sets is $$\sigma$$-locally finite iff it is a countable union of locally finite families)
3. Every open cover of $$X$$ has a locally finite refinement (of any kind).
4. Every open cover of $$X$$ has a locally finite closed refinement.

The following is what I tried.

Denote each closed parcompact subspace $$U_{n}$$. Think about an open covering $$\mathscr{A}$$ of $$X$$. To prove paracompactness, I should prove that $$\mathscr{A}$$ satisfies $$1$$ of the above lemma. However, because $$X$$ is a 'countable' union of $$U_{n}$$, I think that the way to the answer is proving that $$\mathscr{A}$$ satisfies $$2$$.

But I can't proceed any longer.

Let $$\mathcal{U}$$ be an open covering of a regular space $$X=\bigcup_\mathbb{N}X_n$$ which is a union of countably many closed paracompact subspaces $$X_n\subseteq X$$ whose interiors $$(X_n)^\circ$$ cover $$X$$.
For each $$n\in\mathbb{N}$$ the family $$\mathcal{U}_n=\{U\cap X_n\}_{U\in\mathcal{U}}$$ is an open covering of the paracompact $$X_n$$. Thus there is a family $$\mathcal{V}_n'$$ of open subsets of $$X$$ which has the property that $$\{V\cap X_n\}_{V\in\mathcal{V}'_n}$$ is a locally-finite open refinement (in $$X_n)$$ of $$\mathcal{U}_n$$. Write $$\mathcal{V}_n=\{V\cap(X_n)^\circ\}_{V\in\mathcal{V}'_n}$$ to obtain a locally-finite family of open subsets of $$X$$ which covers the interior $$(X_n)^\circ$$ and refines $$\mathcal{U}_n$$.
The collection $$\mathcal{V}=\bigcup_\mathbb{N}\mathcal{V}_n$$ is now a $$\sigma$$-locally-finite open refinement of $$\mathcal{U}$$.
Appeal to part $$2$$ of your theorem to complete.
• My solution is the same as yours, but I search for this problem because it has a star (*) in the book of Munkres. It is good to see that the solution is not hard. Also, I don't know why the author restrict $Xn$ to be closed, as we only need to assume their interior covers X. Commented Dec 7, 2020 at 6:56