Locally Compact Space - Formal Definition

Formal Definition

Let X be a topological space. Most commonly X is called locally compact, if every point of X has a compact neighbourhood.

There are other common definitions: They are all equivalent if X is a Hausdorff space (or preregular). But they are not equivalent in general:

1. every point of X has a compact neighbourhood.

2. every point of X has a closed compact neighbourhood.

2‘. every point of X has a relatively compact neighbourhood.

2‘‘. every point of X has a local base of relatively compact neighbourhoods.

3. every point of X has a local base of compact neighbourhoods.

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.