\section{$\sigma$-Compact Spaces}
\label{section:sigma-compact}

\begin{definition}[$\sigma$-Compact]
\label{definition:sigma-compact}
    Let $X$ be a topological space, then $X$ is \textbf{$\sigma$-compact} if there exits $\seq{K_n} \subset 2^X$ compact such that $X = \bigcup_{n \in \natp}K_n$.
\end{definition}

\begin{definition}[Exhaustion by Compact Sets]
\label{definition:compact-exhaustion}
    Let $X$ be a topological space and $\seq{U_n} \subset 2^X$, then $\seq{U_n}$ is an \textbf{exhaustion of $X$ by compact sets} if:
    \begin{enumerate}
        \item For each $n \in \natp$, $U_n$ is open and precommpact.
        \item For each $n \in \natp$, $\ol{U_n} \subset U_{n+1}$.
    \end{enumerate}
\end{definition}