# Intersection of Open Set and Complement of Compact Set Is Open

compactnessgeneral-topologyparacompactness

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?

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).