Hemicompact space

In mathematics, in the field of topology, a topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set in the sequence. Clearly, this forces the union of the sequence to be the whole space, because every point is compact and hence must lie in one of the compact sets.

Symbolically, a topological space $X$ is said to be hemicompact if there exists a sequence of compact subsets $\{\; K\_n\; \}\_\{n\; in\; mathbb\{N$ such that $K\_n\; subseteq\; mathrm\{int\}(K\_\{n+1\})$ for all $n\; geq\; 1$, $X\; =\; cup\_\{n\; in\; mathbb\{N\; K\_n$, and any compact subset $K\; subseteq\; X$ is contained in some $K\_n$. (Here, $mathrm\{int\}(A)$ denotes the interior of the set $A$.)

Some facts about hemicompactness:

* Every compact space is hemicompact.
* The real line is hemicompact.
* Every first countable hemicompact space is locally compact.
* Every locally compact Lindelof space is hemicompact.

ee also

* Compact space
* Locally compact space
* Lindelof space