In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
and related areas of
mathematics, a metrizable space is a
topological space that is
homeomorphic to a
metric space. That is, a topological space
is said to be metrizable if there is a
metric such that the topology induced by
d is
\mathcal. Metrization theorems are
s that give sufficient condition">theorem">, \infty) such that the topology induced by d is \mathcal. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.
Properties
Metrizable spaces inherit all topological properties from metric spaces. For example, they are
Hausdorff Hausdorff space">Hausdorff spaces (and hence Normal space">normal and Tychonoff space">Tychonoff) and First-countable space">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 mapping, contraction maps than a metric space to which it is homeomorphic.
Metrization theorems
One of the first widely recognized metrization theorems was . 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
Tikhonov in 1926. What
Urysohn
Pavel Samuilovich Urysohn () (February 3, 1898 – August 17, 1924) was a Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of which ar ...
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 collection
In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension.
A collection of subsets of a topological spa ...
s of open sets. For a closely related theorem see the
Bing metrization theorem
In topology, the Bing metrization theorem, named after R. H. Bing, characterizes when a topological space is metrizable.
Formal statement
The theorem states that a topological space X is metrizable if and only if it is regular
The term regu ...
.
Separable metrizable spaces can also be characterized as those spaces which are
homeomorphic to a subspace of the
Hilbert cube
In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, ...
\lbrack 0, 1 \rbrack ^\N, that is, 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
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
. 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.
Examples
The group of unitary operators
\mathbb(\mathcal) on a separable Hilbert space
\mathcal endowed
with the
strong operator topology is metrizable (see Proposition II.1 in
[Neeb, Karl-Hermann, On a theorem of S. Banach. J. Lie Theory 7 (1997), no. 2, 293–300.]).
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
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with ...
, used in
algebraic geometry,
* the
topological vector space of all
functions from the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
\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
Line with two origins, also called the ' is a
non-Hausdorff manifold (and thus cannot be metrizable). Like all manifolds, it is
locally homeomorphic to
Euclidean space and thus
locally metrizable (but not
metrizable) and
locally Hausdorff (but not
Hausdorff). It is also a
T1 locally regular space
In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' can b ...
but not a
semiregular space A semiregular space is a topological space whose regular open sets (sets that equal the interiors of their closures) form a base for the topology.
Examples and sufficient conditions
Every regular space is semiregular, and every topological spa ...
.
The
long line is locally metrizable but not metrizable; in a sense it is "too long".
See also
*
*
*
*
*
* , the property of a topological space of being homeomorphic to a
uniform space, or equivalently the topology being defined by a family of
pseudometrics
References
{{PlanetMath attribution, id=1538, title=Metrizable
General topology
Manifolds
Metric spaces
Properties of topological spaces
Theorems in topology