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.

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:

Logical relations among the conditions:

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 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.

Condition (4) is used, for example, in Bourbaki. In almost all applications, locally compact spaces are indeed also Hausdorff. These locally compact Hausdorff (LCH) spaces are thus the spaces that this article is primarily concerned with.

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:

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.

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