locally compact

A topological spaceMathworldPlanetmath X is locally compact at a point xX if there exists a compact set K which contains a nonempty neighborhoodMathworldPlanetmathPlanetmath U of x. The space X is locally compact if it is locally compact at every point xX.

Note that local compactness at x does not require that x have a neighborhood which is actually compact, since compact open sets are fairly rare and the more relaxed condition turns out to be more useful in practice. However, it is true that a space is locally compact at x if and only if x has a precompact neighborhood.

Title locally compact
Canonical name LocallyCompact
Date of creation 2013-03-22 12:38:24
Last modified on 2013-03-22 12:38:24
Owner djao (24)
Last modified by djao (24)
Numerical id 6
Author djao (24)
Entry type Definition
Classification msc 54D45
Related topic Compact
Defines local compactness