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.

To be precise, a topological space X is locally compact iff every point has a local base of compact neighborhoodss. (Note that these neighborhoods do not have to be open themselves but only need to contain an open set containing the given point.) Other definitions may be found in the literature, as discussed in the section Non-Hausdorff spaces below; however, this is the definition used in Wikipedia. The various definitions of local compactness all coincide for Hausdorff spaces. Almost all locally compact spaces studied in applications are Hausdorff, and this article is thus primarily concerned with locally compact Hausdorff spaces.

Table of contents

1 Examples and counterexamples 1.1 Compact Hausdorff spaces 1.2 Locally compact Hausdorff spaces that are not compact 1.3 Hausdorff spaces that are not locally compact 2 Facts about locally compact Hausdorff spaces 2.4 The point at infinity 2.5 Locally compact groups 3 Non-Hausdorff spaces

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];
• any closed topological manifold;
• 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).

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 Q_p of p-adic numbers is locally compact for any prime number p, 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 Facts below, no Hausdorff space can possibly be locally compact if it isn't also a Tychonoff space; there are some examples of Hausdorff spaces that aren't Tychonoff spaces in that article. But there are also examples of Tychonoff spaces that fail to be locally compact, such as:

• the space Q of rational numbers, since its compact subsets are all finite and don't constitute neighborhoods;
• the subspace {(0,0)} union {(x,y) : x > 0} of R², since the origin doesn't have a compact neighborhood;
• the lower limit topology or upper limit topology on the set R of real numbers (useful in the study of one-sided limits);
• any infinite-dimensional topological vector space, 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 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.

Facts about locally compact Hausdorff spaces

As mentioned in Examples, any compact Hausdorff space is also locally compact, and any locally compact Hausdorff space is in fact a Tychonoff space.

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 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.

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-Cech 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; but if X is locally compact and Hausdorff, then 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 the Gelfand-Naimark theorem. Forming the one-point compactification a(X) of X corresponds under this duality to adjoing 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 locally compact Hausdorff group G carries natural measures called the Haar measures which allow one to integrate functions defined on G. Lebesgue measure on the real line R is a special case of this.

The Pontryagin dual of an abelian topological group A is locally compact iff A is locally compact. More precisely, Pontrjagin 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.

Non-Hausdorff spaces

Much of the theory of locally compact Hausdorff spaces also works for preregular spaces. For example, just as any locally compact Hausdorff space is a Tychonoff space, so any locally compact preregular space is a completely regular 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. The theory of locally compact regular spaces can be derived from the theory of locally compact Hausdorff spaces by considering Kolmogorov equivalence.

The study of local compactness for spaces that aren't even regular is much less developed. In fact, even the definition of "locally compact" is not universally agreed upon. The various definitions include:

• every point has a compact neighbourhood;
• every point has a closed compact neighbourhood;
• every point has a local base of compact neighbourhoods (the definition used in Wikipedia).

All of these definitions are equivalent for Hausdorff (or even preregular) spaces, but only after some time has it become clear that the last definition is the most useful for the general case. However, that general case has not been developed in this article.