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 ...
and related areas of
mathematics, a metrizable space is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
that is
homeomorphic to a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
. That is, a topological space
is said to be metrizable if there is a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
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 paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
spaces (and hence
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
and
Tychonoff) and
first-countable
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 (local base) ...
. 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
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
, for example, may have a different set of
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
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'' ca ...
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 are f ...
had shown, in a paper published posthumously in 1925, was that every second-countable ''
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
'' 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
The Nagata–Smirnov metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space X is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (t ...
, 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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
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
The Nagata–Smirnov metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space X is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (t ...
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 space X ...
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 and T0 and has ...
.
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
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. This topology differs from another, perhaps more natural-s ...
.
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 mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
. 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
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...
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
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
on an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
or on the
spectrum of a ring, used in
algebraic geometry,
* the
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
of all
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s from the
real line \R to itself, with the
topology of pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared.
Definition
Suppose that X is a set and ...
.
The real line with the
lower limit topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of inte ...
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 In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff. ...
, also called the ' is a
non-Hausdorff manifold In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff ...
(and thus cannot be metrizable). Like all manifolds, it is
locally homeomorphic to
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
and thus
locally metrizable (but not
metrizable
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, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
) and
locally Hausdorff (but not
Hausdorff). It is also a
T1 locally regular space 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 space ...
.
The
long line Long line or longline may refer to:
*'' Long Line'', an album by Peter Wolf
* Long line (topology), or Alexandroff line, a topological space
*Long line (telecommunications), a transmission line in a long-distance communications network
*Longline fi ...
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
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
, 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