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
x\inU\subseteqK
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.
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:[1]
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 ,[2] [3] 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[4] calls (2), (2'), (2") strongly locally compact to contrast with property (1), which they call locally compact.
Spaces satisfying condition (4) are exactly the 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
K
U
x
V
x
K\capU
V
Condition (5) is used, for example, in Bourbaki.[5] 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.
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:
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:
\{(0,0)\}\cup((0,infty) x R)
R2
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.
Every locally compact preregular space is, in fact, completely regular.[7] 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.
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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
b(X)
a(X)
a(X)
Intuitively, the extra point in
a(X)
|f(x)|<e
a(X)=X\cup\{infty\}
g(infty)=0.
For a locally compact Hausdorff space X, the set
C0(X)
C0(X)
\R
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.