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...

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...

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

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...

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...

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

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...

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

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...

by the space of continuous functions on a compact Hausdorff space X {\displaystyle X} with values in the real or complex numbers. This space, denoted...

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...

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...

unit square. (Alternatively, we could use the theorem that every compact metric space is a continuous image of the Cantor set to get the function f {\displaystyle...

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....

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...

Eberlein, is a compact topological space homeomorphic to a subset of a Banach space with the weak topology. Every compact metric space, more generally...

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...

locally compact at a point x ∈ X if x lies in the interior of some compact subset of X. X is a locally compact space if it is locally compact at every...

Quasitopological space – Function in topology Relatively compact subspace – Subset of a topological space whose closure is compact Space (mathematics) –...

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...

topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set in...

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

It is associated with the compact-open topology. Let ( X , T ) {\displaystyle (X,{\mathcal {T}})} be a topological space and ( Y , d Y ) {\displaystyle...

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...

compactness, since a compact subset of a Hausdorff space is closed. Thus, every compact Hausdorff space is H-closed. The notion of an H-closed space has...