In
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
and related areas of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a metrizable space is a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
that is
homeomorphic to a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
. That is, a topological space
is said to be metrizable if there is a
metric such that the topology induced by
d is
\tau. ''Metrization theorems'' are
s that give sufficient condition">theorem">, \infty) such that the topology induced by d is \tau. ''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 and Tychonoff space">Tychonoff) and First-countable space">first-countable. However, some properties of the metric, such as Complete metric space">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 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
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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
A collection of subsets of a topological space X is said to be locally finite if each point in the space has a neighbourhood that intersects only finitely many of the sets in the collection.
In the mathematical field of topology, local finiteness ...
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 \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 (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
. 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.]).
Non-normal spaces cannot be metrizable; important examples include
* the
Zariski topology on an
algebraic variety or on the
spectrum of a ring, used in
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
,
* the
topological vector space of all
functions from the
real line \R to itself, with the
topology of pointwise convergence.
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 \mathbb, the set of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of in ...
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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
and thus
locally metrizable (but not metrizable) and
locally Hausdorff (but not
Hausdorff). It is also a
T1 locally regular space 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
Topological spaces