Supercompact space

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In mathematics, in the field of topology, a topological space is called supercompact if there is a subbasis such that every open cover of the topological space from elements of the subbasis has a subcover with just two subbasis elements. By the Alexander subbase lemma, every supercompact space is compact.

The following are typical examples of supercompact spaces:

  • Linearly ordered compact spaces
  • Compact metrizable spaces

Some important facts about supercompactness:

  • A product of supercompact spaces is supercompact
  • Every supercompact space is compact
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