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. When locally compact spaces are Hausdorff they are called locally compact Hausdorff, which are of particular interest in mathematical analysis.
Formal definition 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 x ∈ U ⊆ K {\displaystyle x\in U\subseteq K} . There are other common definitions, which are all equivalent if X is a Hausdorff space (or preregular), but 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. 4. every point of X has a local base of closed compact neighbourhoods. 5. 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 called weakly locally compact, 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. Steen & Seebach calls (2), (2'), (2") strongly locally compact to contrast with property (1), which they call locally compact. Spaces satisfying condition (4) are exactly the locally compact regular spaces. Indeed, such a space is regular, as every point has a local base of closed neighbourhoods. Conversely, in a regular locally compact space suppose a point x {\displaystyle x} has a compact neighbourhood K {\displaystyle K} . By regularity, given an arbitrary neighbourhood U {\displaystyle U} of x {\displaystyle x} , there is a closed neighbourhood V {\displaystyle V} of x {\displaystyle x} contained in K ∩ U {\displaystyle K\cap U} and V {\displaystyle V} is compact as a closed set in a compact set. Condition (5) is used, for example, in Bourbaki. Any space that is locally compact (in the sense of condition (1)) and also Hausdorff automatically satisfies all the conditions above. Since in most applications locally compact spaces are also Hausdorff, these locally compact Hausdorff spaces will thus be the spaces that this article is primarily concerned with.
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:
the unit interval [0,1]; the Cantor set; the Hilbert cube.
Locally compact Hausdorff spaces that are not compact The Euclidean spaces Rn (and in particular the real line R) are locally compact as a consequence of the Heine–Borel theorem. Topological manifolds share the local properties of Euclidean spaces and are therefore also all locally compact. This even includes nonparacompact manifolds such as the long line. All discrete spaces are locally compact and Hausdorff (they are just the zero-dimensional manifolds). These are compact only if they are finite. All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology. This provides several examples of locally compact subsets of Euclidean spaces, such as the unit disc (either the open or closed version). The space Qp of p-adic numbers is locally compact, because it is homeomorphic to the Cantor set minus one point. Thus locally compact spaces are as useful in p-adic analysis as in classical analysis.
Hausdorff spaces that are not locally compact As mentioned in the following section, if a Hausdorff space is locally compact, then it is also a Tychonoff space. For this reason, examples of Hausdorff spaces that fail to be locally compact because they are not Tychonoff spaces can be found in the article dedicated to Tychonoff spaces. But there are also examples of Tychonoff spaces that fail to be locally compact, such as:
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