Properties
Every first countable hemicompact space is locally compact.
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 (the existence of such a sequence follows from the hemicompactness of ). Denote
for and . Then
defines a metric on which induces the compact-open topology.
Read more about this topic: Hemicompact Space
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—John Locke (1632–1704)
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