Definition 3.11.2 (Exhaustion by Compact Sets).label Let $X$ be a topological space and $\curl{U_n}_{1}^{\infty}\suf \cali{P}(X)$, then $\curl{U_n }_{1}^{\infty}$ is an exhaustion of $X$ by compact sets if:

1. (1)

   For each $n\in\N^{+}$, $U_{n}$ is open and precompact

2. (2)

   For each $n\in\N^{+}$, $\cl{U_n}\suf U_{n+1}$