%CE%A3-compact Space
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%CE%A3-compact Space

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

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

Properties and examples

  • Every compact space is ?-compact, and every ?-compact space is Lindelöf (i.e. every open cover has a countable subcover).[3] The reverse implications do not hold, for example, standard Euclidean space (Rn) is ?-compact but not compact,[4] and the lower limit topology on the real line is Lindelöf but not ?-compact.[5] In fact, the countable complement topology on any uncountable set is Lindelöf but neither ?-compact nor locally compact.[6] However, it is true that any locally compact Lindelöf space is ?-compact.
  • A Hausdorff, Baire space that is also ?-compact, must be locally compact at at least one point.
  • 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 ?-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, ?-compactness implies local compactness.
  • The previous property implies for instance that R? is not ?-compact: if it were ?-compact, it would necessarily be locally compact since R? is a topological group that is also a Baire space.
  • Every hemicompact space is ?-compact.[7] The converse, however, is not true;[8] for example, the space of rationals, with the usual topology, is ?-compact but not hemicompact.
  • The product of a finite number of ?-compact spaces is ?-compact. However the product of an infinite number of ?-compact spaces may fail to be ?-compact.[9]
  • A ?-compact space X is second category (respectively Baire) if and only if the set of points at which is X is locally compact is nonempty (respectively dense) in X.[10]

See also

Notes

  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

  • Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970). ISBN 0-03-079485-4.
  • Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

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