A

topological space is said to be

locally compact if every

point has a

compact neighborhood. Local compactness is a useful property to require of

geometric objects because it is purely

topological (i.e., does not require a

metric or

uniform structure) but implies many of the same properties we are used to in

metric contexts. In particular, a

locally compact Hausdorff space is

completely regular and a version of the

Baire category theorem holds in

locally compact spaces.

Euclidean space **R**^{n} is locally compact, and therefore so is any finite-dimensional manifold (not only smooth manifolds, but also topological manifolds and PL-manifolds). Harmonic analysis takes place almost entirely on locally compact topological groups and their homogenous spaces. Note however that one should not expect useful function spaces to be locally compact, because a locally compact normed linear space is finite-dimensional, and nearly all function spaces encountered in analysis are not.