Definition 5.17.2 (Exhaustion by Compact Sets).label Let $X$ be a topological space and $\seq{U_n}\subset 2^{X}$, then $\seq{U_n}$ is an exhaustion of $X$ by compact sets if:

1. (1)

   For each $n \in \natp$, $U_{n}$ is open and precommpact.

2. (2)

   For each $n \in \natp$, $\ol{U_n}\subset U_{n+1}$.