A topological space$X$ is called hemicompact iff there exists a sequence $(K_n)_{n\in\mathbb{N}}$ of compact subsets $K_n\subseteq X$ such that $X=\bigcup_{n\in\mathbb{N}} K_n$ and $K_n \subseteq int(K_{n+1})$ holds.

Properties

If $X$ is hemicompact and $(K_n)$ a sequence as in the definition, then every compact subset $K\subseteq X$ is contained in one of the $K_n$. In other words: $\lbrace K_n \mid n\in\mathbb{N}\rbrace$ is cofinal in the set of all compact subsets (partially ordered by inclusion)