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 (resp. Baire) if and only if the set of points at which is X is locally compact is nonempty (resp. dense) in X.