σ-compact space

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In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.[1]

A space is said to be σ-locally compact if it is both σ-compact and locally compact.[2]

Properties and examples[edit]

  • If G is a topological group and G is locally compact at one point, then G is locally compact everywhere. Therefore, the previous property tells us that if G is a σ-compact, Hausdorff topological group that is also a Baire space, then G is locally compact. This shows that for Hausdorff topological groups that are also Baire spaces, σ-compactness implies local compactness.
  • The previous property implies for instance that Rω is not σ-compact: if it were σ-compact, it would necessarily be locally compact since Rω is a topological group that is also a Baire space.
  • Every hemicompact space is σ-compact.[7] The converse, however, is not true;[8] for example, the space of rationals, with the usual topology, is σ-compact but not hemicompact.
  • The product of a finite number of σ-compact spaces is σ-compact. However the product of an infinite number of σ-compact spaces may fail to be σ-compact.[9]
  • A σ-compact space X is second countable (resp. Baire) if and only if the set of points at which is X is locally compact is nonempty (resp. dense) in X.[10]

See also[edit]

Notes[edit]

  1. ^ Steen, p.19; Willard, p. 126.
  2. ^ Steen, p. 21.
  3. ^ Steen, p. 19.
  4. ^ Steen, p. 56.
  5. ^ Steen, p. 75–76.
  6. ^ Steen, p. 50.
  7. ^ Willard, p. 126.
  8. ^ Willard, p. 126.
  9. ^ Willard, p. 126.
  10. ^ Willard, p. 188.

References[edit]