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 locally compact. That terminology can be somewhat confusing as it does not fit the usual pattern of σ- meaning a countable union of spaces satisfying ; that's why such spaces are more commonly referred to explicitly as σ-compact 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. The reverse implications do not hold, for example, standard Euclidean space 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.
- 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 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. 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 if and only if the set of points at which is X is locally compact is nonempty in X.