Concept
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. More precisely, it is a topological space in which every point has a compact neighborhood. In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces. Let X be a topological space. Most commonly X is called locally compact if every point x of X has a compact neighbourhood, i.e., there exists an open set U and a compact set K, such that . There are other common definitions: They are all equivalent if X is a Hausdorff space (or preregular). But they are not equivalent in general:
- every point of X has a compact neighbourhood.
- 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.
- every point of X has a local base of compact neighbourhoods.
- every point of X has a local base of closed compact neighbourhoods.
- X is Hausdorff and satisfies any (or equivalently, all) of the previous conditions. Logical relations among the conditions: Each condition implies (1). Conditions (2), (2′), (2′′) are equivalent. Neither of conditions (2), (3) implies the other. Condition (4) implies (2) and (3). Compactness implies conditions (1) and (2), but not (3) or (4). 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. Spaces satisfying (1) are also occasionally called , as they satisfy the weakest of the conditions here. As they are defined in terms of relatively compact sets, spaces satisfying (2), (2'), (2") can more specifically be called locally relatively compact.
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