# σ-compact space

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

- 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 (
**R**^{n}) 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 on any uncountable set 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, 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
**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.[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 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

- Steen, p.19; Willard, p. 126.
- Steen, p. 21.
- Steen, p. 19.
- Steen, p. 56.
- Steen, p. 75–76.
- Steen, p. 50.
- Willard, p. 126.
- Willard, p. 126.
- Willard, p. 126.
- 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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