Locally compact space

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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.

Contents

Formal definition

Let X be a topological space. The following are common definitions for X is locally compact, and are equivalent if X is a Hausdorff space (or preregular). They are not equivalent in general:

Logical relations among the conditions:

  • Conditions (2), (2‘), (2‘‘) are equivalent.
  • Neither of conditions (2), (3) implies the other.
  • Each condition implies (1).
  • Compactness implies conditions (1) and (2), but not (3).

Condition (1) is probably the most commonly used definition, since it is the least restrictive and the others are equivalent to it when X is Hausdorff. This equivalence is a consequence of the facts that compact subsets of Hausdorff spaces are closed, and closed subsets of compact spaces are compact.

Authors such as Munkres and Kelley use the first definition. Willard uses the third. In Steen and Seebach, a space which satisfies (1) is said to be locally compact, while a space satisfying (2) is said to be strongly locally compact.

In almost all applications, locally compact spaces are also Hausdorff, and this article is thus primarily concerned with locally compact Hausdorff (LCH) spaces.

Examples and counterexamples

Compact Hausdorff spaces

Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article compact space. Here we mention only:

Locally compact Hausdorff spaces that are not compact

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