nLab
locally compact space

Definition

A locally compact space is a space such that every point has a neighborhood base consisting of compact sets. 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.

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 $X$ arises in this way, since it can be considered an open subspace in its one-point compactification $X\bigsqcup \{\mathrm{\infty }\}$ (where the open neighborhoods of the adjoined point $\mathrm{\infty }$ are precisely those of the form $K^c\bigsqcup \{\mathrm{\infty }\}$, where $K^c$ is the complement of a compact subset $K\subseteq X$).

3. The reals, complexes, and $𝔭$-adic completions of algebraic number field?s (with respect to a prime ideal $𝔭$ in the ring of integers) are locally compact. In characteristic $p$, the field of Laurent series ${𝔽}_q((t))$ over a finite field with $q$ 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 field?s.

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 ${ℝ}^n$, 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.

Basic category-theoretic results

Perhaps the most important consequence of local compactness for categorical topology is that locally compact spaces are exponentiable, i.e., if $Y$ is locally compact, then $Y\times -:\mathrm{Top}\to \mathrm{Top}$ has a right adjoint $(-{)}^Y:\mathrm{Top}\to \mathrm{Top}$. In fact, this is almost an abstract definition of local compactness: for $T_0$ 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 $\mathrm{Top}$. 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 $(X_p,K_p{)}_{p\in P}$ be a collection of pairs of spaces where each $X_p$ is locally compact and $K_p\subseteq X_p$ is a compact open subspace. The restricted direct product of the collection is the colimit of the filtered diagram consisting of spaces

$$D_F=\prod _{p\in F}{X}_p\times \prod _{p\notin F}{K}_p$$D_F = \prod_{p \in F} X_p \times \prod_{p \notin F} K_p

where $F$ ranges over all finite subsets of $P$, together with inclusions $D_F\subseteq D_{F\prime }$ where $F\subseteq F\prime$. We observe that each of the $D_F$ 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; the main examples of such restricted direct products, as many readers will be aware, include topologized adele ring?s and idele group?s. In the case of adele rings, the collection of pairs is $(K_{𝔭},O_{𝔭})$ where $K_{𝔭}$ is the $𝔭$-adic completion of a number field? $K$ and $O_{𝔭}$ is the $𝔭$-adic completion of the ring of integers $O\subseteq K$.

Locally compact spaces are closed under coproducts in $\mathrm{Top}$. 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 $\mathrm{Top}$ of diagrams of locally compact spaces.

Further properties

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