Metrization theorem
Encyclopedia
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
is said to be metrizable if there is a metric
such that the topology induced by d is
. Metrization theorems are theorem
s that give sufficient conditions for a topological space to be metrizable.
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.
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.)
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
s 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
, 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 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.
The long line
is locally metrizable but not metrizable; in a sense it is "too long".
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and related areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a metrizable space is a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
that is homeomorphic
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
to a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
. That is, a topological space
is said to be metrizable if there is a metricsuch that the topology induced by d is
. Metrization theorems are theoremTheorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
s 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 HausdorffHausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
paracompact spaces (and hence normal
Normal space
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
and Tychonoff
Tychonoff space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....
) and first-countable
First-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis...
. 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
Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
, for example, may have a different set of contraction maps
Contraction mapping
In mathematics, a contraction mapping, or contraction, on a metric space is a function f from M to itself, with the property that there is some nonnegative real number k...
than a metric space to which it is homeomorphic.
Metrization theorems
The first really useful metrization theorem was Urysohn's metrization theorem. This states that every Hausdorff second-countable regular spaceRegular space
In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...
is metrizable. So, for example, every second-countable manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
is metrizable. (Historical note: The form of the theorem shown here was in fact proved by Tychonoff
Andrey Nikolayevich Tychonoff
Andrey Nikolayevich Tikhonov was a Soviet and Russian mathematician known for important contributions to topology, functional analysis, mathematical physics, and ill-posed problems. He was also inventor of magnetotellurics method in geology. Tikhonov originally published in German, whence the...
in 1926. What Urysohn
Pavel Samuilovich Urysohn
Pavel Samuilovich Urysohn, Pavel Uryson was a Jewish mathematician who is best known for his contributions in the theory of dimension, and for developing Urysohn's Metrization Theorem and Urysohn's Lemma, both of which are fundamental results in topology...
had shown, in a paper published posthumously in 1925, was that every second-countable normal
Normal space
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
Hausdorff space is metrizable.)
Several other metrization theorems follow as simple corollaries to Urysohn's Theorem. For example, a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
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
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....
s of open sets. For a closely related theorem see the Bing metrization theorem
Bing metrization theorem
In topology, the Bing metrization theorem, named after R. H. Bing, characterizes when a topological space is metrizable. The theorem states that a topological space X is metrizable if and only if it is regular and T0 and has a σ-discrete basis...
.
Separable metrizable spaces can also be characterized as those spaces which are homeomorphic to a subspace of the Hilbert cube
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...
, i.e. the countably infinite product of the unit interval (with its natural subspace topology from the reals) with itself, endowed with the product topologyProduct topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
.
A space is said to be locally metrizable if every point has a metrizable neighbourhood
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
. 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 of non-metrizable spaces
Non-normal spaces cannot be metrizable; important examples include- the Zariski topologyZariski topologyIn algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
on an algebraic varietyAlgebraic varietyIn mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
or on the spectrum of a ringSpectrum of a ringIn abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
, used in algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, - the topological vector spaceTopological vector spaceIn mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
of all functionFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s from the real lineReal lineIn mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R to itself, with the topology of pointwise convergence. - the Strong operator topologyStrong operator topologyIn functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the weakest locally convex topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number \|Tx\| is continuous for each vector...
on the set of unitary operators on a Hilbert SpaceHilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
(often denoted 
)
The real line with the lower limit topology
Lower limit topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties...
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.
The long line
Long line (topology)
In topology, the long line is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore it serves as one of the basic counterexamples of topology...
is locally metrizable but not metrizable; in a sense it is "too long".
See also
- Uniformizability, the property of a topological space of being homeomorphic to a uniform spaceUniform spaceIn the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
, or equivalently the topology being defined by a family of pseudometricPseudometricPseudometric may refer to:* Pseudo-Riemannian manifold* Pseudometric space...
s - Moore space (topology)Moore space (topology)In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. Equivalently, a topological space X is a Moore space if the following conditions hold:...