Disjoint Compact Sets are Separable from each other in a Hausdorff Space

Last updated Nov 1, 2022

Let $X$ be a Hausdorff Topological Space. Then if $K, L \subset X$ are Compact and $K \cap L = \emptyset$, then there exists $U, V \subset X$ Open so that

1. $K \subset U$
2. $L \subset V$
3. $U \cap V = \emptyset$

TODO Sketch: This uses Compact Sets are Separable from Points in a Hausdorff Space to create an Open Cover for $L$. We can basically run the exact same logic from that proof. $\square$