Σ-compact space explained

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 (weakly) locally compact.^[2] 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.^[3]

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).^[4] The reverse implications do not hold, for example, standard Euclidean space (Rⁿ) is σ-compact but not compact,^[5] and the lower limit topology on the real line is Lindelöf but not σ-compact.^[6] In fact, the countable complement topology on any uncountable set is Lindelöf but neither σ-compact nor locally compact.^[7] However, it is true that any locally compact Lindelöf space is σ-compact.
• (The irrational numbers)

R\setminus Q is not σ-compact.^[8]

• 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.^[9] The converse, however, is not true;^[10] 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.^[11]
• 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.^[12]

References

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

Notes and References

1. Steen, p. 19; Willard, p. 126.
2. Steen, p. 21.
3. Web site: A question about local compactness and $\sigma$-compactness . Mathematics Stack Exchange . en.
4. Steen, p. 19.
5. Steen, p. 56.
6. Steen, p. 75 - 76.
7. Steen, p. 50.
8. Book: Hart . K.P. . Nagata . J. . Vaughan . J.E. . Encyclopedia of General Topology . 2004 . Elsevier . 0 444 50355 2 . 170.
9. Willard, p. 126.
10. Willard, p. 126.
11. Willard, p. 126.
12. Willard, p. 188.