Hemicompact space

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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.[1] 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[edit]

Properties[edit]

Every hemicompact space is σ-compact and if in addition it is first countable then it is locally compact.

Applications[edit]

If is a hemicompact space, then the space of all continuous functions to a metric space with the compact-open topology is metrizable.[2] 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 ). Define psedometrics

Then

defines a metric on which induces the compact-open topology.

See also[edit]

Notes[edit]

  1. ^ Willard 2004, Problem set in section 17.
  2. ^ Conway 1990, Example IV.2.2.

References[edit]

  • Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
  • Conway, J. B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. 96. Springer Verlag. ISBN 0-387-97245-5.