3.11 $\s$-Compact Spaces

Definition 3.11.1 ($\s$-Compact).label Let $X$ be a topological space, then $X$ is $\s$-compact if there exists $\curl{K_n}_{1}^{\infty}\suf \cali{P}(X)$ compact such that $X=\cups{}{n\in \N^+}K_{n}$

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