In
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
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
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 points ...
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
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 settin ...
. 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 mathema ...
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
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 ...
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
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'' 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 collections 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
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
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, c ...
\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-seemin ...
.
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; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
. 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
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. Mo ...
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
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 Y ...
.
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 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
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 à ...
to
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
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 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