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.

Formal definition

Let X be a topological space. Most commonly X is called locally compact, if every point of X has a compact neighbourhood.

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.

Logical relations among the conditions:

• Conditions (2), (2′), (2″) are equivalent.
• Neither of conditions (2), (3) implies the other.
• Each condition implies (1).
• Compactness implies conditions (1) and (2), but not (3).

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 a 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.

Authors such as Munkres and Kelley use the first definition. Willard uses the third. In Steen and Seebach, a space which satisfies (1) is said to be locally compact, while a space satisfying (2) is said to be strongly locally compact.

Warning: Allen Hatcher uses another definition! On page 62 of his Algebraic Topology, he explains that if he says locally [something] then he means that every open neighbourhood of any point contains a [something] type of neighbourhood for the point. Thus, for Hatcher, locally compact has a stronger meaning. Interestingly, this stronger definition allows for us to omit the Hausdorff condition in many situations. For example, all of Hatcher's locally compact spaces have the property that the functor: ${\displaystyle (-)\times X}$ has the right adjoint :${\displaystyle (-)^{X}}$.

By definition of a local base, this property is identical to

3'. for every point x of X the compact neighbourhoods of x form a local base.

and therefore equivalent to (3).

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:

Hausdorff spaces that are not locally compact

As mentioned in the following section, no Hausdorff space can possibly be locally compact if it is not also a Tychonoff space; there are some examples of Hausdorff spaces that are not Tychonoff spaces in that article. But there are also examples of Tychonoff spaces that fail to be locally compact, such as:

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 numbers Q is compact and therefore locally compact in senses (1) and (2) but it is not locally compact in sense (3).
• The particular point topology on any infinite set is locally compact in sense (3) but not in sense (2), because it has no nonempty closed compact subspaces containing the particular point. The same holds for the real line with the upper topology.

Properties

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 Hausdorff space is a Baire space. That is, the conclusion of the Baire category theorem holds: the interior of every union of countably many nowhere dense subsets is empty.

A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X can be written as the set-theoretic difference of two closed subsets of Y. As a corollary, a dense subspace X of a locally compact Hausdorff space Y is locally compact if and only if X is an open subset of Y. Furthermore, if a subspace X of any Hausdorff space Y is locally compact, then X still must be the difference of two closed subsets of Y, although the converse needn't hold in this case.

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 locally compact spaces local uniform convergence is the same as compact convergence.

The point at infinity

Since every locally compact Hausdorff space X is Tychonoff, it can be embedded in a compact Hausdorff space b(X) 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 a(X) with just one extra point. (The one-point compactification can be applied to other spaces, but a(X) 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 a(X) 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 |f(x)| < e 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 a(X) = X ∪ {∞} where g(∞) = 0.

The set C0(X) of all continuous complex-valued functions that vanish at infinity is a C* algebra. In fact, every commutative C* algebra is isomorphic to C0(X) for some unique (up to homeomorphism) locally compact Hausdorff space X. More precisely, the categories of locally compact Hausdorff spaces and of commutative C* algebras are dual; this is shown using the Gelfand representation. Forming the one-point compactification a(X) of X corresponds under this duality to adjoining an identity element to C0(X).

Locally compact groups

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 R 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.

References

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