More precisely, it is a topological space in which every point has a compact neighborhood.
When locally compact spaces are Hausdorff they are called locally compact Hausdorff, which are of particular interest in mathematical analysis.
There are other common definitions: They are all equivalent if X is a Hausdorff space (or preregular).
As they are defined in terms of relatively compact sets, spaces satisfying (2), (2'), (2") can more specifically be called locally relatively compact.
[5][6] Steen & Seebach[7] calls (2), (2'), (2") strongly locally compact to contrast with property (1), which they call locally compact.
[8][2] 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
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.
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.
[13] 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.
[14][15] That is, the conclusion of the Baire category theorem holds: the interior of every countable union of nowhere dense subsets is empty.
Without the Hausdorff hypothesis, some of these results break down with weaker notions of locally compact.
Every closed set in a weakly locally compact space (= condition (1) in the definitions above) is weakly locally compact.
as an open set which is not weakly locally compact.
This section explores compactifications of locally compact spaces.
So to avoid trivialities it is assumed below that the space X is not compact.
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
whenever the point x lies outside of K. This definition makes sense for any topological space X.
For a locally compact Hausdorff space X, the set
of all continuous complex-valued functions on X that vanish at infinity is a commutative C*-algebra.
for some unique (up to homeomorphism) locally compact Hausdorff space X.
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.