Are nice spaces paracompact iff their connected components are?

Question

Suppose a space $X$ is the coproduct of its connected components. Is it paracompact if and only if they all are?

Answer 1

Yes. Let $\mathscr{C}$ be the set of connected components, each of which is clopen in $X$. If $\mathscr{U}$ is any open cover of $X$, $\mathscr{V}=\{U\cap C:U\in\mathscr{U}\text{ and }C\in\mathscr{C}\}$ is an open refinement of $\mathscr{U}$ and of $\mathscr{C}$. For $C\in\mathscr{C}$ let $\mathscr{V}_C=\{V\in\mathscr{V}:V\cap C\ne\varnothing\}$. If each $C\in\mathscr{C}$ is paracompact, each $\mathscr{V}_C$ has a locally finite open refinement $\mathscr{R}_C$ covering $C$, and $\bigcup_{C\in\mathscr{C}}\mathscr{R}_C$ is then a locally finite open refinement of $\mathscr{U}$.

Conversely, if $X$ is paracompact, then so is every closed subspace, and in particular so is each $C\in\mathscr{C}$.

Answer 2

Yes: in general, a coproduct of spaces $X=\coprod X_i$ is paracompact iff each $X_i$ is paracompact. For the forward direction, any closed subset of a paracompact space is paracompact. For the reverse direction, given an open cover of $X$, you can intersect each of the sets in the cover with each $X_i$ to get a refinement where each open set is contained in some $X_i$. If the $X_i$ are paracompact, you can then choose a locally finite refinement of the sets contained in each $X_i$, and these together will give a locally finite refinement of your original cover of $X$.