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.

Examples and nonexamples

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.

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Here we mention only:

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• the unit interval ;
• 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.

Related Topics:
Euclidean space - Real line - Heine-Borel theorem

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Topological manifolds share the local properties of Euclidean spaces and are therefore also all locally compact.

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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).

Related Topics:
Nonparacompact - Long line - Discrete space - Zero

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All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology.

Related Topics:
Open - Closed subset - Subspace

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This provides several examples of locally compact subsets of Euclidean spaces, such as the unit disc (either the open or closed version).

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The space Qp of p-adic numbers is locally compact for any prime number p, because it is homeomorphic to the Cantor set minus one point.

Related Topics:
''p''-adic numbers - Prime number - Homeomorphic - Cantor set

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Thus locally compact spaces are as useful in p-adic analysis as in classical analysis.

Related Topics:
''p''-adic analysis - Analysis

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Hausdorff spaces that are not locally compact

As mentioned in the following section, 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.

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But there are also examples of Tychonoff spaces that fail to be locally compact, such as:

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• the space Q of rational numbers, since its compact subsets all have empty interior and therefore 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 T₀, hence Hausdorff, Topological vector space, which is infinite-dimensional, 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.

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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).

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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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