σ-compact space

In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.<ref>Steen, p. 19; Willard, p. 126.</ref>

A space is said to be σ-locally compact if it is both σ-compact and (weakly) locally compact.<ref>Steen, p. 21.</ref> 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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).<ref>Steen, p. 19.</ref> The reverse implications do not hold, for example, standard Euclidean space (Rⁿ) is σ-compact but not compact,<ref>Steen, p. 56.</ref> and the lower limit topology on the real line is Lindelöf but not σ-compact.<ref>Steen, p. 75–76.</ref> In fact, the countable complement topology on any uncountable set is Lindelöf but neither σ-compact nor locally compact.<ref>Steen, p. 50.</ref> However, it is true that any locally compact Lindelöf space is σ-compact.
• (The irrational numbers) <math>\mathbb R\setminus\mathbb Q</math> is not σ-compact.<ref>Template:Cite book</ref>
• 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.<ref>Willard, p. 126.</ref> The converse, however, is not true;<ref>Willard, p. 126.</ref> 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.<ref>Willard, p. 126.</ref>
• 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.<ref>Willard, p. 188.</ref>

See also

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References

• Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970). Template:ISBN.
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