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.

Facts about locally compact Hausdorff spaces

As mentioned in the previous section, any compact Hausdorff space is also locally compact, and any locally compact Hausdorff space is in fact a Tychonoff space.

Related Topics:
Compact - Tychonoff space

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Every locally compact Hausdorff space is a Baire space.

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That is, the conclusion of the Baire category theorem holds: the interior of every union of countably many nowhere dense subsets is empty.

Related Topics:
Baire category theorem - Interior - Union - Countably many - Nowhere dense - Subset - Empty

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A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X can be written as the set-theoretic difference of two closed subsets of Y.

Related Topics:
Subspace - If and only if - Set-theoretic difference - Closed - Subset

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As a corollary, a dense subspace X of a compact Hausdorff space Y is locally compact if and only if X is an open subset of Y.

Related Topics:
Dense - Open subset

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Furthermore, if a subspace X of any Hausdorff space Y is locally compact, then X still must be the difference of two closed subsets of Y, although the converse needn't hold in this case.

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Quotient spaces of locally compact Hausdorff spaces are compactly generated.

Related Topics:
Quotient space - Compactly generated

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Conversely, every compactly generated Hausdorff space is a quotient of some locally compact Hausdorff space.

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The point at infinity

Since every locally compact Hausdorff space X is Tychonoff, it can be embedded in a compact Hausdorff space b(X) using the Stone-Čech compactification.

Related Topics:
Embedded - Stone-Čech compactification

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But in fact, there is a simpler method available in the locally compact case; the one-point compactification will embed X in a compact Hausdorff space a(X) with just one extra point.

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(The one-point compactification can be applied to other spaces, but a(X) will be Hausdorff if and only if X is locally compact and Hausdorff.)

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The locally compact Hausdorff spaces can thus be characterised as the open subsets of compact Hausdorff spaces.

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Intuitively, the extra point in a(X) can be thought of as a point at infinity.

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The point at infinity should be thought of as lying outside every compact subset of X.

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Many intuitive notions about tendency towards infinity can be formulated in locally compact Hausdorff spaces using this idea.

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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 |f(x)| < e whenever the point x lies outside of K.

Related Topics:
Continuous - Real - Complex - Function - Domain - Vanish at infinity - Positive number - Point

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This definition makes sense for any topological space X; but if X is locally compact and Hausdorff, then the set C0(X) of all continuous complex-valued functions that vanish at infinity is a C* algebra.

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In fact, every commutative C* algebra is isomorphic to C0(X) for some unique (up to homeomorphism) locally compact Hausdorff space X.

Related Topics:
Commutative - Isomorphic - Unique - Up to - Homeomorphism

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More precisely, the categories of locally compact Hausdorff spaces and of commutative C* algebras are dual; this is shown using the Gelfand representation.

Related Topics:
Categories - Dual - Gelfand representation

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Forming the one-point compactification a(X) of X corresponds under this duality to adjoining an identity element to C0(X).

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Locally compact groups

The notion of local compactness is important in the study of topological groups mainly because every locally compact Hausdorff group G carries natural measures called the Haar measures which allow one to integrate functions defined on G.

Related Topics:
Topological group - Measures - Haar measure - Integrate

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Lebesgue measure on the real line R is a special case of this.

Related Topics:
Lebesgue measure - Real line

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The Pontryagin dual of an topological abelian group A is locally compact iff A is locally compact.

Related Topics:
Pontryagin dual - Topological abelian group - Iff

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More precisely, Pontrjagin duality defines a self-duality of the category of locally compact Abelian groups.

Related Topics:
Duality - Category

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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.

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