Σ-compact space

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.
• Moreover, every σ-compact space is Lindelöf (i.e. every open cover has a countable subcover).
• The reverse implications of the previous two examples 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 or compact. In fact, the countable complement topology is Lindelof but neither σ-compact nor locally compact.
• Let X be a Hausdorff, Baire space that is also σ-compact. Then X 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 topological group that is also a Baire space, then G is locally compact. This shows that for topological groups that are also Baire spaces, σ-compactness implies local compactness
• We can conclude from the previous property that R^ω is not σ-compact (if it were σ-compact, it would locally compact since R^ω is a topological group that is also a Baire space)

See also

• Exhaustion by compact sets
• Locally compact space
• Lindelöf space