Metrizable space

Properties

Metrizable spaces inherit all topological properties from metric spaces. For example, they are

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

Urysohn's Theorem can be restated as: A topological space is separable 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 metrizable if 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 many locally finite collections of open sets. For a closely related theorem see the Bing metrization theorem.

Separable metrizable spaces can also be characterized as those spaces which are

${\displaystyle \lbrack 0,1\rbrack ^{\mathbb {N} },}$ that is, the countably infinite product of the unit interval (with its natural subspace topology from the reals) with itself, endowed with the

A space is said to be locally metrizable if every point has a metrizable

Examples

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

Examples of non-metrizable spaces

Non-normal spaces cannot be metrizable; important examples include

• the Zariski topology on an algebraic variety or on the spectrum of a ring, used in algebraic geometry,
• the
  real line
  ${\displaystyle \mathbb {R} }$ to itself, with the
  topology of pointwise convergence
  .

The real line with the lower 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.

Locally metrizable but not metrizable

The

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

See also

• Apollonian metric – Romanian mathematician and poet
• Bing metrization theorem – Characterizes when a topological space is metrizable
• Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
• Moore space (topology) – developable regular Hausdorff spacePages displaying wikidata descriptions as a fallback
• 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 a uniform space, or equivalently the topology being defined by a family of pseudometrics