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.Examples
Every hemicompact space is σ-compact and if in addition it is first countable then it is locally compact.Applications
If is a hemicompact space, then the space of all continuous functions to a metric space with the compact-open topology is metrizable. To see this, take a sequence of compact subsets of such that every compact subset of lies inside some compact set in this sequence. Define pseudometrics
Then
defines a metric on which induces the compact-open topology.
