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.

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