areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space ( X , τ ) {\displaystyle...

In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (X, T) for which there exists at least...

homeomorphic to a uniform space (equipped with the topology induced by the uniform structure). Any (pseudo)metrizable space is uniformizable since the...

metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is...

on the class of metrizable spaces. Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense...

topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable...

the term "Banach space" and Banach in turn then coined the term "Fréchet space" to mean a complete metrizable topological vector space, without the local...

of a paracompact space and a compact space is always paracompact. Every metric space is paracompact. A topological space is metrizable if and only if it...

generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm. Metrizable topologies on vector spaces have been studied...

all metrizable spaces are normal, all metric spaces are Moore spaces. Moore spaces are a lot like regular spaces and different from normal spaces in the...

in pseudometrizable spaces. In a first countable T1 space, every singleton is a Gδ set. A subspace of a completely metrizable space X {\displaystyle X}...

collection of compact spaces is compact. (This is Tychonoff's theorem, which is equivalent to the axiom of choice.) In a metrizable space, a subset is compact...

A metrizable space is an AR if and only if it is contractible and an ANR. By Dugundji, every locally convex metrizable topological vector space V {\textstyle...

Hausdorff topological space is compact metrizable if and only if it is a continuous image of a Cantor space. Let C(X) denote the space of all real-valued...

completely metrizable spaces, spaces for which there exists at least one complete metric inducing the given topology. Completely metrizable spaces can be...

Locally metrizable/Locally metrisable A space is locally metrizable if every point has a metrizable neighbourhood. Locally path-connected A space is locally...

continuous surjective map from a compact metrizable space to an Hausdorff space, then Y is compact metrizable. The last fact follows from f(X) being compact...

sequential. Thus every metrizable or pseudometrizable space — in particular, every second-countable space, metric space, or discrete space — is sequential....

pseudometric space is a Baire space. In particular, every completely metrizable topological space is a Baire space. (BCT2) Every locally compact regular space is...

spaces (and hence all metrizable spaces) are perfectly normal Hausdorff; All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal...

it is closed and bounded. In fact, general topology tells us that a metrizable space is compact if and only if it is sequentially compact, so that the Bolzano–Weierstrass...

completely metrizable space is completely metrizable. Every open subspace of a Baire space is a Baire space. Every closed subspace of a compact space is compact...

0. {\displaystyle y=0.} Every metrizable topological space is pseudometrizable. A pseudometrizable space is metrizable if and only if it is Hausdorff...

Hawaiian earring is a one-dimensional, compact, locally path-connected metrizable space. Although H {\displaystyle \mathbb {H} } is locally homeomorphic to...

compact, first-countable spaces that are not compact (these are necessarily not metrizable spaces). One such space is the ordinal space [ 0 , ω 1 ) . {\displaystyle...

analysis, DF-spaces, also written (DF)-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological...

cardinality. Every metrizable space is paracompact, hence completely uniformizable. As there exist metrizable spaces that are not completely metrizable, complete...

example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated...

non-metrizable spaces like βN, the Stone–Čech compactification of the integers. Certain hyperspaces, measure spaces, and probabilistic metric spaces turn...

topological space to be a Baire space. (BCT1) Every complete pseudometric space is a Baire space. In particular, every completely metrizable topological space is...