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...

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

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...

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...

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...

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

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...

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...

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

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...

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

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...

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}...

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...

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

continuous surjective map from a compact metrizable space to an Hausdorff space, then Y {\displaystyle Y} is compact metrizable. The last fact follows from f (...

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...

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

X} is metrizable. This applies in particular to separable reflexive Banach spaces. Although the weak topology of the unit ball is not metrizable in general...

hedgehog space where K is the cardinality of the continuum. Kowalsky's theorem, named after Hans-Joachim Kowalsky, states that any metrizable space of weight...

space be metrizable in a manner that satisfies the above properties. All Banach spaces and Fréchet spaces are F-spaces. In particular, a Banach space...

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

not every topological space can be given a metric. Topological spaces which are compatible with a metric are called metrizable and are particularly well-behaved...

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

first-countable space is a Fréchet–Urysohn space. Consequently, every second-countable space, every metrizable space, and every pseudometrizable space is a Fréchet–Urysohn...

second-countable is replaced by metrizable. These two formulations are equivalent. In one direction a compact Hausdorff space is a normal space and, by the Urysohn...

C_{c}^{k}(U)} is not metrizable and thus also not normable (see this footnote for an explanation of how the non-metrizable space C c k ( U ) {\displaystyle...

conditions of Definition 3. Every metrizable space is monotonically normal. Every linearly ordered topological space (LOTS) is monotonically normal. This...