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.
Properties.
Every hemicompact space is σ-compact and if in addition it is first countable then it is locally compact.
Applications.
If formula_1 is a hemicompact space, then the space formula_2 of all continuous functions formula_3 to a metric space formula_4 with the compact-open topology is metrizable. To see this, take a sequence formula_5 of compact subsets of formula_1 such that every compact subset of formula_1 lies inside some compact set in this sequence (the existence of such a sequence follows from the hemicompactness of formula_1). Define pseudometrics
Then
defines a metric on formula_11 which induces the compact-open topology.