compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has...

topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely...

X} . Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (if...

space is said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact if it is both σ-compact...

a topological space X {\displaystyle X} is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made...

ambient space). The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These...

example of a countably compact space that is not compact. Every compact space is countably compact. A countably compact space is compact if and only if it...

by Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only...

of a compact topological space is relatively compact (since a closed subset of a compact space is compact). And in an arbitrary topological space every...

commonly used notion of compactness, which requires the existence of a finite subcover. A hereditarily Lindelöf space is a topological space such that every subspace...

contain them Compact operator, a linear operator that takes bounded subsets to relatively compact subsets, in functional analysis Compact space, a topological...

continuous. Functions with compact support on a topological space X {\displaystyle X} are those whose closed support is a compact subset of X . {\displaystyle...

related branches of mathematics, a core-compact topological space X {\displaystyle X} is a topological space whose partially ordered set of open subsets...

with domain a compact metric space (Dunford & Schwartz 1958, p. 382). Modern formulations of the theorem allow for the domain to be compact Hausdorff and...

facts: Every compact space is feebly compact. Every feebly compact paracompact space is compact.[citation needed] Every feebly compact space is pseudocompact...

In mathematics, a topological space X {\displaystyle X} is said to be limit point compact or weakly countably compact if every infinite subset of X {\displaystyle...

{\displaystyle \ X\ ,} but a compact ball / neighborhood exists if and only if   X   {\displaystyle \ X\ } is a finite-dimensional vector space. In particular, no...

Compactness Every product of compact spaces is compact (Tychonoff's theorem). A product of locally compact spaces need not be locally compact. However, an arbitrary...

compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators...

result of making a topological space into a compact space. A compact space is a space in which every open cover of the space contains a finite subcover....

subcover. Every compact space is Lindelöf and paracompact. Therefore, every compact Hausdorff space is normal. See also quasicompact. Compact-open topology...

setting of metric spaces. Other notions, such as continuity, compactness, and open and closed sets, can be defined for metric spaces, but also in the even...

Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named...

a compact space is compact. A compact subset of a Hausdorff space is closed. Every continuous bijection from a compact space to a Hausdorff space is...

L^{p}} spaces can be generalized. Many of the results of finite group representation theory are proved by averaging over the group. For compact groups...

given topological space ordered by inclusion is a complete Heyting algebra. Compact space – Type of mathematical space Convergence space – Generalization...

the Cure ended in 2005. Bamonte was one of three members in the band Compact Space, along with Christian Eigner and Florian Kraemmer. In the band, Bamonte...

representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is...

Normed space – Vector space on which a distance is definedPages displaying short descriptions of redirect targets Locally compact field Locally compact group –...

extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician...