Locally compact space Totally Explained

Everything about Local Compactness totally explained

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.

Formal definition

Let X be a topological space. The following are common definitions for X is locally compact, and are equivalent if X is a Hausdorff space (or preregular). They are not equivalent in general:

1. every point of X has a compact neighbourhood.
2. every point of X has a closed compact neighbourhood.
2‘. every point has a relatively compact neighbourhood.
2‘‘. every point has a local base of relatively compact neighbourhoods.
3. every point of X has a local base of compact neighbourhoods.

Logical relations among the conditions:

Conditions (2), (2‘), (2‘‘) are equivalent.

Neither of conditions (2), (3) implies the other.

Each condition implies (1).

Compactness implies conditions (1) and (2), but not (3). Condition (1) is probably the most commonly used definition, since it's 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.
Authors such as Munkres and Kelley use the first definition. Willard uses the third. In Steen and Seebach, a space which satisfies (1) is said to be locally compact, while a space satisfying (2) is said to be strongly locally compact.
In almost all applications, locally compact spaces are also Hausdorff, and this article is thus primarily concerned with locally compact Hausdorff (LCH) spaces.

Examples and counterexamples

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. Here we mention only:

the unit interval [0,1];

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.

Topological manifolds share the local properties of Euclidean spaces and are therefore also all locally compact. 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). These are compact only if they're finite.

All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology. This provides several examples of locally compact subsets of Euclidean spaces, such as the unit disc (either the open or closed version).

The space Q_p of p-adic numbers is locally compact, because it's homeomorphic to the Cantor set minus one point. Thus locally compact spaces are as useful in p-adic analysis as in classical analysis.

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

the space Q of rational numbers, since its compact subsets all have empty interior and therefore are not neighborhoods;

the subspace where g(∞) = 0.
The set C₀(X) of all continuous complex-valued functions that vanish at infinity is a C* algebra. In fact, every commutative C* algebra is isomorphic to C₀(X) for some unique (up to homeomorphism) locally compact Hausdorff space X. More precisely, the categories of locally compact Hausdorff spaces and of commutative C* algebras are dual; this is shown using the Gelfand representation. Forming the one-point compactification a(X) of X corresponds under this duality to adjoining an identity element to C₀(X).

Locally compact groups

The notion of local compactness is important in the study of topological groups mainly because every Hausdorff locally compact group G carries natural measures called the Haar measures which allow one to integrate functions defined on G. Lebesgue measure on the real line R is a special case of this.
The Pontryagin dual of a topological abelian group A is locally compact if and only if A is locally compact. 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.

Further Information

Get more info on 'Local Compactness'.

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