locally compact space

As a locally compact space, the real line can be compactified in several different ways.

For locally compact spaces an integration theory is then recovered.

A product of locally compact spaces need not be locally compact.

Every locally compact space is compactly generated.

However, the last chapter of the book addresses limitations, especially for use in probability theory, of the restriction to locally compact spaces.

For example, all of Hatcher's locally compact spaces have the property that the functor: has the right adjoint :.

Let be a complex, separable Hilbert space, a locally compact space and a measure on .

It is currently able to characterise the category of (not necessarily Hausdorff) computably based locally compact spaces.

For locally compact spaces local uniform convergence is the same as compact convergence.

One-point compactification extends this definition to locally compact spaces without basepoints: .

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

The Borel Moore homology theory applies to general locally compact spaces, and is closely related to sheaf theory.

If X is a locally compact space, assumed Hausdorff the family of all compact subsets satisfies the further conditions, making it paracompactifying.

Verdier duality was introduced by as an analog for locally compact spaces of the coherent duality for schemes due to Grothendieck.

This category includes all CW complexes, locally compact spaces, and first-countable spaces (such as metric spaces).

However, an arbitrary product of locally compact spaces where all but finitely many are compact is locally compact (This condition is sufficient and necessary).

In mathematics, Borel Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by .

In mathematics, the Riesz-Markov-Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures.

Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982.

In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions f on either metric space or locally compact space is continuous.

The algebra of all bounded continuous real- or complex-valued functions on some locally compact space (again with pointwise operations and supremum norm) is a Banach algebra.

To complete the buildup of measure theory for locally compact spaces from the functional-analytic viewpoint, it is necessary to extend measure (integral) from compactly supported continuous functions.

The theory of Radon measures has most of the good properties of the usual theory for locally compact spaces, but applies to all Hausdorff topological spaces.

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.

Consequently, it is a locally compact space which is a union of a countable number of compact subsets, a separable space, a paracompact and completely regular space.