Σ compact space

Updated on Jan 19, 2024

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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.

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

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 (Rⁿ) 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 is Lindelöf but neither σ-compact nor locally 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 (resp. Baire) if and only if the set of points at which is X is locally compact is nonempty (resp. dense) in X.

References

Σ-compact space Wikipedia

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