4.17 $\sigma$-Compact Spaces

Definition 4.17.1 ($\sigma$-Compact). Let $X$ be a topological space, then $X$ is $\sigma$-compact if there exits $\seq{K_n}\subset 2^{X}$ compact such that $X = \bigcup_{n \in \natp}K_{n}$.

Definition 4.17.2 (Exhaustion by Compact Sets). 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. For each $n \in \natp$, $U_{n}$ is open and precommpact.

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