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.[1]
Formal definition
Let X be a
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 compactneighbourhood.
- 2″. every point of X has a local baseof 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:[2]
- 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,[3][4] 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.[5][6] Steen & Seebach[7] 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.[8][2] 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 has a compact neighbourhood . By regularity, given an arbitrary neighbourhood of , there is a closed neighbourhood of contained in and is compact as a closed set in a compact set.
Condition (5) is used, for example, in Bourbaki.[9] 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 (LCH) 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 real line R) are locally compact as a consequence of the Heine–Borel theorem.
- nonparacompact manifolds such as the long line.
- All zero-dimensional manifolds). These are compact only if they are finite.
- All unit disc(either the open or closed version).
- The space Qp of 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:
- the space Q of rational numbers (endowed with the topology from R), since any neighborhood contains a Cauchy sequence corresponding to an irrational number, which has no convergent subsequence in Q;
- the subspace of , since the origin does not have a compact neighborhood;
- the upper limit topology on the set R of real numbers (useful in the study of one-sided limits);
- any T0, hence Hausdorff, topological vector space that is infinite-dimensional, such as an infinite-dimensional Hilbert space.
The first two examples show that a subset of a locally compact space need not be locally compact, which contrasts with the open and closed subsets in the previous section. The last example contrasts with the Euclidean spaces in the previous section; to be more specific, a Hausdorff topological vector space is locally compact if and only if it is finite-dimensional (in which case it is a Euclidean space). This example also contrasts with the Hilbert cube as an example of a compact space; there is no contradiction because the cube cannot be a neighbourhood of any point in Hilbert space.
Non-Hausdorff examples
- The one-point compactification of the rational numbersQ 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 topologyon 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 topologyon 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 topologyon 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).
General classes of examples
- Every space with an Alexandrov topology is locally compact in senses (1) and (3).[10]
Properties
Every locally compact
Every locally compact regular space, in particular every locally compact Hausdorff space, is a Baire space.[14][15] That is, the conclusion of the
A
Without the Hausdorff hypothesis, some of these results break down with weaker notions of locally compact. Every closed set in a
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,
The point at infinity
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
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
Gelfand representation
For a locally compact Hausdorff space X, the set of all continuous complex-valued functions on X that vanish at infinity is a commutative
Locally compact groups
The notion of local compactness is important in the study of
defined on G. TheThe
See also
- Compact group – Topological group with compact topology
- F. Riesz's theorem
- Locally compact field
- Locally compact quantum group – relatively new C*-algebraic approach toward quantum groups
- Locally compact group – topological group G for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be defined
- σ-compact space – union of countably many compact topological spaces
- Core-compact space
Citations
- ^ Folland 1999, p. 131, Sec. 4.5.
- ^ (PDF) from the original on 2015-09-10.
- ., p. 3
- ISBN 978-3-540-22264-4.
- Zbl 0522.54003
- arXiv:2002.05943 [math.GN].
- ^ Steen & Seebach, p. 20
- ^ Kelley 1975, ch. 5, Theorem 17, p. 146.
- ISBN 3-540-19374-X.
- ].Theorem 5
- ^ Schechter 1996, 17.14(d), p. 460.
- ^ "general topology - Locally compact preregular spaces are completely regular". Mathematics Stack Exchange.
- ^ Willard 1970, theorem 19.3, p.136.
- ^ Kelley 1975, Theorem 34, p. 200.
- ^ Schechter 1996, Theorem 20.18, p. 538.
References
- ISBN 978-0-471-31716-6.
- ISBN 978-0387901251.
- ISBN 978-0131816299.
- OCLC 175294365.
- MR 0507446.
- Willard, Stephen (1970). General Topology. ISBN 978-0486434797.