Intersection of Open Set and Complement of Compact Set Is Open

I am reviewing the blog post: https://amathew.wordpress.com/2010/08/17/paracompactness. Under Lemma 7, the author states that each $U_{i+1} - \overline{U_{i-2}}$ is open in $X$ (which is locally compact, $\sigma$-compact and Hausdorff). The nested increasing sequences sequence of sets $\{U_i\}$ and $\{\overline{U_i}\}$ satisfy $U_i \subseteq U_{i+1}$, $\overline{U_i} \subseteq \overline{U_{i+1}}$, $\overline{U_i} \subseteq U_{i+1}$, and each $\overline{U_i}$ is compact.

I am struggling to understand why $U_{i+1} - \overline{U_{i-2}}$ is open. If $\overline{U_{i-2}}$ were closed, then we could apply the fact that the intersection of two open sets is open. However, $\overline{U_i}$ is not necessarily closed. Am I missing something?

Solution 1:

If $O$ is open and $C$ is closed in $X$, $O - C = O \cap (X - C)$ which is is the intersection of two open sets (the complement of a closed set is open, after all) and so open.

This applies to $U_{i+1} - \overline{U_{i-2}}$ in particular: for any set $S$ the set $\overline{S}$ is closed (it's not called the closure for nothing).