Metrization theorem

In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space ( X , T ) {displaystyle (X,{mathcal {T}})} is said to be metrizable if there is a metric In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space ( X , T ) {displaystyle (X,{mathcal {T}})} is said to be metrizable if there is a metric such that the topology induced by d is T {displaystyle {mathcal {T}}} . Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable. Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first-countable. However, some properties of the metric, such as completeness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable uniform space, for example, may have a different set of contraction maps than a metric space to which it is homeomorphic. One of the first widely recognized metrization theorems was Urysohn's metrization theorem. This states that every Hausdorff second-countable regular space is metrizable. So, for example, every second-countable manifold is metrizable. (Historical note: The form of the theorem shown here was in fact proved by Tychonoff in 1926. What Urysohn had shown, in a paper published posthumously in 1925, was that every second-countable normal 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. The Nagata–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, 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 homeomorphic to a subspace of the Hilbert cube [ 0 , 1 ] N {displaystyle lbrack 0,1 brack ^{mathbb {N} }} , i.e. the countably infinite product of the unit interval (with its natural subspace topology from the reals) with itself, endowed with the product topology. A space is said to be locally metrizable if every point has a metrizable neighbourhood. Smirnov proved that a locally metrizable space is metrizable if and only if it is Hausdorff and paracompact. In particular, a manifold is metrizable if and only if it is paracompact. The group of unitary operators U ( H ) {displaystyle mathbb {U} ({mathcal {H}})} on a separable Hilbert space H {displaystyle {mathcal {H}}} endowedwith the strong operator topology is metrizable (see Proposition II.1 in ).