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...

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...

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...

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...

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...

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...

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

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

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

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...

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...

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...

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...

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....

Any vector space (including those that are infinite dimensional) endowed with the trivial topology is a compact (and thus locally compact) complete pseudometrizable...

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...

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...

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...

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

%5B%5BWikipedia%3ARedirects+for+discussion%5D%5D+debate+closed+as+delete #REDIRECT Σ-compact space From ASCII-only title: This is a redirect from a title in only basic...

the data into a compact latent representation. VAEs are known for their ability to generate new data samples from the learned latent space. Multimodality...

tight if and only if it is sequentially weakly compact. If X {\displaystyle X} is a metrizable compact space, then every collection of (possibly complex)...

priori constraints) on how the space in which we live is connected or whether it wraps around on itself as a compact space. Though certain cosmological...