Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

Related Topics:
Topology - Mathematics - Topological space - Compact space

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To be precise, a topological space X is locally compact iff every point has a local base of compact neighborhoods.

Related Topics:
Iff - Point - Local base - Neighborhood

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(Note that these neighborhoods do not have to be open themselves but need only contain an open set containing the given point.)

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Other definitions may be found in the literature, as discussed in the section Non-Hausdorff spaces below; however, this is the definition used in Wikipedia.

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The various definitions of local compactness all coincide for Hausdorff spaces.

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Almost all locally compact spaces studied in applications are Hausdorff, and this article is thus primarily concerned with locally compact Hausdorff spaces.

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