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.
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 Rω is not σ-compact: if it were σ-compact, it would necessarily be locally compact since Rω 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]
See also
Exhaustion by compact sets – in analysis, a sequence of compact sets that converges on a given setPages displaying wikidata descriptions as a fallback