Metrizable space information

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...

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

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...

Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which...

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...

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...

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

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

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...

property coinciding with completely metrizability on the class of metrizable spaces, but including some non-metrizable spaces as well), or that it is completely...

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

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

including those that are not metrizable or Hausdorff. Completeness is an extremely important property for a topological vector space to possess. The notions...

Hausdorff paracompact space is collectionwise normal. In particular, every metrizable space is collectionwise normal. Note: The Hausdorff condition is necessary...

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

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

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...

spaces, countable compactness and sequential compactness are equivalent. More generally, the same holds for sequential spaces. For metrizable spaces,...