In
topology and related areas of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a metrizable space is a
topological space that is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
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
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
regular space is metrizable. So, for example, every second-countable
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
is metrizable. (Historical note: The form of the theorem shown here was in fact proved by
Tikhonov 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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to a subspace of the
Hilbert cube \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. 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 the ...
, used in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
,
* 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 (geometry), 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 ...
\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
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
If f : X \to Y is a local homeomorphism, X is said to be an � ...
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.
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