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Intopology and related areas ofmathematics, ametrizable space is atopological space that ishomeomorphic to ametric space. That is, a topological space${\displaystyle (X,\tau )}$ is said to be metrizable if there is ametric${\displaystyle d:X\times X\to [0,\mathrm{\infty })}$ such that the topology induced by${\displaystyle d}$ is${\displaystyle \tau .}$^[1][2]Metrization theorems aretheorems that givesufficient conditions for a topological space to be metrizable.

Metrizable spaces inherit all topological properties from metric spaces. For example, they areHausdorffparacompact spaces (and hencenormal andTychonoff) andfirst-countable. However, some properties of the metric, such ascompleteness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizableuniform space, for example, may have a different set ofcontraction maps than a metric space to which it is homeomorphic.

One of the first widely recognized metrization theorems wasUrysohn's metrization theorem. This states that every Hausdorffsecond-countableregular space is metrizable. So, for example, every second-countablemanifold is metrizable. (Historical note: The form of the theorem shown here was in fact proved byTikhonov in 1926. WhatUrysohn had shown, in a paper published posthumously in 1925, was that every second-countablenormal Hausdorff space is metrizable.) The converse does not hold: there exist metric spaces that are not second countable, for example, an uncountable set endowed with the discrete metric.^[3] TheNagata–Smirnov metrization theorem, described below, provides a more specific theorem where the converse does hold.

Several other metrization theorems follow as simple corollaries to Urysohn's theorem. For example, acompact Hausdorff space is metrizable if and only if it is second-countable.

Urysohn's Theorem can be restated as: A topological space isseparable and metrizable if and only if it is regular, Hausdorff and second-countable. The Nagata–Smirnov metrization theorem extends this to the non-separable case. It states that a topological space is metrizableif and only if it is regular, Hausdorff and has a σ-locally finite base. A σ-locally finite base is a base which is a union of countably manylocally finite collections of open sets. For a closely related theorem see theBing metrization theorem.

Separable metrizable spaces can also be characterized as those spaces which arehomeomorphic to a subspace of theHilbert cube${\displaystyle [0,1{]}^{\mathbb{N}},}$ that is, the countably infinite product of the unit interval (with its natural subspace topology from the reals) with itself, endowed with theproduct topology.

A space is said to belocally metrizable if every point has a metrizableneighbourhood. Smirnov proved that a locally metrizable space is metrizable if and only if it is Hausdorff andparacompact. In particular, a manifold is metrizable if and only if it is paracompact.

The group of unitary operators${\displaystyle \mathbb{U}(\mathcal{H})}$ on a separable Hilbert space${\displaystyle \mathcal{H}}$ endowedwith thestrong operator topology is metrizable (see Proposition II.1 in^[4]).

Non-normal spaces cannot be metrizable; important examples include

• theZariski topology on analgebraic variety or on thespectrum of a ring, used inalgebraic geometry,
• thetopological vector space of allfunctions from thereal line${\displaystyle \mathbb{R}}$ to itself, with thetopology of pointwise convergence.

The real line with thelower limit topology is not metrizable. The usual distance function is not a metric on this space because the topology it determines is the usual topology, not the lower limit topology. This space is Hausdorff, paracompact and first countable.

TheLine with two origins, also called thebug-eyed line is anon-Hausdorff manifold (and thus cannot be metrizable). Like all manifolds, it islocally homeomorphic toEuclidean space and thuslocally metrizable (but not metrizable) andlocally Hausdorff (but notHausdorff). It is also aT₁locally regular space but not asemiregular space.

Thelong line is locally metrizable but not metrizable; in a sense, it is "too long".

• Apollonian metric – Romanian mathematician and poet (1895 - 1961)
• Bing metrization theorem – Characterizes when a topological space is metrizable
• Metrizable topological vector space – Topological vector space whose topology can be defined by a metric
• Moore space (topology)
• Nagata–Smirnov metrization theorem – Characterizes when a topological space is metrizable
• Uniformizability – Topological space whose topology is generated by a uniform structurePages displaying short descriptions of redirect targets, the property of a topological space of being homeomorphic to auniform space, or equivalently the topology being defined by a family ofpseudometrics

This article incorporates material from Metrizable onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

• General topology
• Manifolds
• Metric spaces
• Properties of topological spaces
• Theorems in topology
• Topological spaces

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