σ-compact Space

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

A space is said to be ''σ''-locally compact if it is both ''σ''-compact and (weakly) locally compact. That terminology can be somewhat confusing as it does not fit the usual pattern of σ-(property) meaning a countable union of spaces satisfying (property); that's why such spaces are more commonly referred to explicitly as ''σ-compact (weakly) locally compact'', which is also equivalent to being exhaustible by compact sets.

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). The reverse implications do not hold, for example, standard Euclidean space (R^''n'') is ''σ''-compact but not compact, and the lower limit topology on the real line is Lindelöf but not ''σ''-compact. In fact, the countable complement topology on any uncountable set is Lindelöf but neither ''σ''-compact nor locally compact. However, it is true that any locally compact Lindelöf space is ''σ''-compact.
*(The irrational numbers) $\mathbb R\setminus\mathbb Q$ is not ''σ''-compact.
*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. The converse, however, is not true; 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.
* 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''.Willard, p. 188.

See also

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Notes

References

* Steen, Lynn A. and Seebach, J. Arthur Jr.; '' Counterexamples in Topology'', Holt, Rinehart and Winston (1970). .
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{{DEFAULTSORT:Compact Space
Compactness (mathematics)
General topology
Properties of topological spaces