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]

Contents 1 Properties and Examples 2 See also 3 Notes 4 References

[edit] Properties and Examples

• Every compact space is σ-compact, and every σ-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 σ-compact but not compact,^[4] and the lower limit topology on the real line is Lindelöf but not σ-compact.^[5] In fact, the countable complement topology is Lindelöf but neither σ-compact nor locally compact.^[6]

• A Hausdorff, Baire space that is also σ-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 σ-compact topological group that is also a Baire space, then G is locally compact. This shows that for topological groups that are also Baire spaces, σ-compactness implies local compactness.

• The previous property implies for instance that R^ω is not σ-compact: if it were σ-compact, it would necessarily be locally compact since R^ω is a topological group that is also a Baire space.

• Every hemicompact space is σ-compact, but the converse is not true.^[7]

• 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.^[8]

[edit] See also

• Exhaustion by compact sets
• Locally compact space
• Lindelöf space

[edit] Notes

[edit] References

• Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970). ISBN 0030794854.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.