nLab
locally compact topological space

Locally compact spaces

Definition

A topological space is locally compact if every point has a neighborhood base? consisting of compact subspaces. It may be considered as an example of a nice topological space.

Note: as observed in the discussion at compact space, many authors choose to include the Hausdorff condition as a matter of course, calling locally compact not-necessarily-Hausdorff spaces ‘locally quasi-compact’. We will not follow that convention here, but the reader should be warned that without the Hausdorff hypothesis, there are several inequivalent notions of local compactness in the literature; see the English Wikipedia for a survey and counterexamples.

A locally compact Hausdorff space may also be called a local compactum; compare compactum.

Examples

  1. Clearly, any discrete space is locally compact.

  2. An open subspace of a compact Hausdorff space is locally compact. In fact, every locally compact Haudorff space XX arises in this way, since it can be considered an open subspace in its one-point compactification X{}X \sqcup \{\infty\} (where the open neighborhoods of the adjoined point \infty are precisely those of the form K c{}K^c \sqcup \{\infty\}, where K cK^c is the complement of a compact subset KXK \subseteq X).

  3. The reals, complexes, and 𝔭\mathfrak{p}-adic completions of algebraic number fields (with respect to a prime ideal 𝔭\mathfrak{p} in the ring of integers) are locally compact. In characteristic pp, the field of Laurent series 𝔽 q((t))\mathbb{F}_q((t)) over a finite field with qq elements, topologized with respect to a discrete valuation, is locally compact. In fact, any non-discrete locally compact field must be of one of these types; they are called local fields.

  4. Finite products of locally compact spaces are locally compact. Closed subspaces of locally compact spaces are locally compact. (Hence locally compact spaces form a finitely complete category.)

  5. Topological manifolds (including “pathological examples” like long lines), being locally homeomorphic to n\mathbb{R}^n, are locally compact.

  6. The only Hausdorff topological vector spaces that are locally compact are finite-dimensional Euclidean spaces. More generally, a TVS is locally compact if and only if its Hausdorff quotient has finite dimension.

Properties

Category-theoretic properties

Perhaps the most important consequence of local compactness for categorical topology is that locally compact Hausdorff spaces are exponentiable, i.e., if YY is locally compact Hausdorff, then Y×:TopTopY \times -: Top \to Top has a right adjoint () Y:TopTop(-)^Y: Top \to Top. In fact, this is almost an abstract definition of local compactness: for T 0T_0 spaces, local compactness is equivalent to being exponentiable. This situation generalises to locales: a result of Hyland is that locale is locally compact if and only if it is exponentiable. (See exponential law for spaces for more details.)

As noted above, locally compact spaces form a finitely complete full subcategory of TopTop. It is not true that arbitrary products of locally compact spaces are locally compact. However, some important examples of locally compact spaces are constructed as restricted direct products, as follows.

Let (X p,K p) pP(X_p, K_p)_{p \in P} be a collection of pairs of spaces where each X pX_p is locally compact and K pX pK_p \subseteq X_p is a compact open subspace. The restricted direct product of the collection is the colimit of the filtered diagram consisting of spaces

D F= pFX p× pFK pD_F = \prod_{p \in F} X_p \times \prod_{p \notin F} K_p

where FF ranges over all finite subsets of PP, together with inclusions D FD FD_F \subseteq D_{F'} where FFF \subseteq F'. We observe that each of the D FD_F is locally compact, and that a filtered colimit or union of a system of open inclusions of locally compact spaces is again locally compact. Therefore, restricted direct products are locally compact, under the hypotheses stated above.

These hypotheses are of course pretty severe; important examples of such restricted direct products include topologized adele rings and idele groups. In the case of adele rings, the collection of pairs is (K 𝔭,O 𝔭)(K_{\mathfrak{p}}, O_{\mathfrak{p}}) where K 𝔭K_{\mathfrak{p}} is the 𝔭\mathfrak{p}-adic completion of a number field KK and O 𝔭O_{\mathfrak{p}} is the 𝔭\mathfrak{p}-adic completion of the ring of integers OKO \subseteq K.

In any event, the category of locally compact spaces does not admit general infinite products. If it did, then so would the category of locally compact Hausdorff spaces, and so would the category of locally compact Hausdorff abelian groups. However, there is no product of countably many copies of the real numbers in LCHAbLCHAb, for if there were, then by utilizing the universal property of the product, it would become a Hausdorff TVS over the real numbers, in contradiction to the fact that the only locally compact Hausdorff TVS are finite-dimensional.

Locally compact spaces are closed under coproducts in TopTop. They do not admit many types of colimits generally; in some sense this is a raison d'être for compactly generated spaces: they are precisely the colimits in TopTop of diagrams of locally compact spaces.

Gelfand duality

Under Gelfand duality the category of (locally) compact Hausdorff topological spaces is equivalent to the opposite category of (non-unital) commutative C-star algebras.

Further properties

Locally compact Hausdorff spaces are paracompact whenever they are also second-countable.

Revised on June 22, 2014 13:23:13 by Toby Bartels (98.16.175.187)