Жергілікті ықшамдық кеңістіктер

Type of topological space in mathematics
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.

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.

The one point compactification of the rational numbers Q is compact and therefore locally compact in senses (1) and (2) but it is not locally compact in senses (3) or (4). The particular point topology on any infinite set is locally compact in senses (1) and (3) but not in senses (2) or (4), because the closure of any neighborhood is the entire space, which is non compact. The disjoint union of the above two examples is locally compact in sense (1) but not in senses (2), (3) or (4). The right order topology on the real line is locally compact in senses (1) and (3) but not in senses (2) or (4), because the closure of any neighborhood is the entire non compact space. The Sierpiński space is locally compact in senses (1), (2) and (3), and compact as well, but it is not Hausdorff or regular (or even preregular) so it is not locally compact in senses (4) or (5). The disjoint union of countably many copies of Sierpiński space is a non compact space which is still locally compact in senses (1), (2) and (3), but not (4) or (5). More generally, the excluded point topology is locally compact in senses (1), (2) and (3), and compact, but not locally compact in senses (4) or (5). The cofinite topology on an infinite set is locally compact in senses (1), (2), and (3), and compact as well, but it is not Hausdorff or regular so it is not locally compact in senses (4) or (5). The indiscrete topology on a set with at least two elements is locally compact in senses (1), (2), (3), and (4), and compact as well, but it is not Hausdorff so it is not locally compact in sense (5).

Every space with an Alexandrov topology is locally compact in senses (1) and (3).

Every locally compact preregular space is, in fact, completely regular. It follows that every locally compact Hausdorff space is a Tychonoff space. Since straight regularity is a more familiar condition than either preregularity (which is usually weaker) or complete regularity (which is usually stronger), locally compact preregular spaces are normally referred to in the mathematical literature as locally compact regular spaces. Similarly locally compact Tychonoff spaces are usually just referred to as locally compact Hausdorff spaces. Every locally compact regular space, in particular every locally compact Hausdorff space, is a Baire space. That is, the conclusion of the Baire category theorem holds: the interior of every countable union of nowhere dense subsets is empty. A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X is locally closed in Y (that is, X can be written as the set theoretic difference of two closed subsets of Y). In particular, every closed set and every open set in a locally compact Hausdorff space is locally compact. Also, as a corollary, a dense subspace X of a locally compact Hausdorff space Y is locally compact if and only if X is open in Y. Furthermore, if a subspace X of any Hausdorff space Y is locally compact, then X still must be locally closed in Y, although the converse does not hold in general. Without the Hausdorff hypothesis, some of these results break down with weaker notions of locally compact. Every closed set in a weakly locally compact space (= condition (1) in the definitions above) is weakly locally compact. But not every open set in a weakly locally compact space is weakly locally compact. For example, the one point compactification of the rational numbers is compact, and hence weakly locally compact. But it contains as an open set which is not weakly locally compact. Quotient spaces of locally compact Hausdorff spaces are compactly generated. Conversely, every compactly generated Hausdorff space is a quotient of some locally compact Hausdorff space. For functions defined on a locally compact space, local uniform convergence is the same as compact convergence.

This section explores compactifications of locally compact spaces. Every compact space is its own compactification. So to avoid trivialities it is assumed below that the space X is not compact. Since every locally compact Hausdorff space X is Tychonoff, it can be embedded in a compact Hausdorff space using the Stone–Čech compactification. But in fact, there is a simpler method available in the locally compact case; the one point compactification will embed X in a compact Hausdorff space with just one extra point. (The one point compactification can be applied to other spaces, but will be Hausdorff if and only if X is locally compact and Hausdorff.) The locally compact Hausdorff spaces can thus be characterised as the open subsets of compact Hausdorff spaces. Intuitively, the extra point in can be thought of as a point at infinity. The point at infinity should be thought of as lying outside every compact subset of X. Many intuitive notions about tendency towards infinity can be formulated in locally compact Hausdorff spaces using this idea. For example, a continuous real or complex valued function f with domain X is said to vanish at infinity if, given any positive number e, there is a compact subset K of X such that whenever the point x lies outside of K. This definition makes sense for any topological space X. If X is locally compact and Hausdorff, such functions are precisely those extendable to a continuous function g on its one point compactification where

For a locally compact Hausdorff space X, the set of all continuous complex valued functions on X that vanish at infinity is a commutative C* algebra. In fact, every commutative C* algebra is isomorphic to for some unique (up to homeomorphism) locally compact Hausdorff space X. This is shown using the Gelfand representation.

The notion of local compactness is important in the study of topological groups mainly because every Hausdorff locally compact group G carries natural measures called the Haar measures which allow one to integrate measurable functions defined on G.
The Lebesgue measure on the real line is a special case of this. The Pontryagin dual of a topological abelian group A is locally compact if and only if A is locally compact. More precisely, Pontryagin duality defines a self duality of the category of locally compact abelian groups. The study of locally compact abelian groups is the foundation of harmonic analysis, a field that has since spread to non abelian locally compact groups.