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 Rn 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.