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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.

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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.

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  • $\begingroup$ 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. $\endgroup$ Commented Dec 7, 2020 at 6:56

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