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In
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 pseudometric space is a
generalization A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common character ...
of 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 setti ...
in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by
Đuro Kurepa Đuro Kurepa (Serbian Cyrillic: Ђуро Курепа, ; 16 August 1907 – 2 November 1993) was a Yugoslav mathematician. Throughout his life, Kurepa published over 700 articles, books, papers, and reviews and over 1,000 scientific reviews. He l ...
in 1934. In the same way as every
normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
is 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 setti ...
, every
seminormed space In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
is a pseudometric space. Because of this analogy the term semimetric space (which has a different meaning 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 ...
) is sometimes used as a synonym, especially in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
. When a topology is generated using a family of pseudometrics, the space is called a gauge space.


Definition

A pseudometric space (X,d) is a set X together with a non-negative real-valued function d : X \times X \longrightarrow \R_, called a , such that for every x, y, z \in X, #d(x,x) = 0. #''Symmetry'': d(x,y) = d(y,x) #''
Subadditivity In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
''/''
Triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
'': d(x,z) \leq d(x,y) + d(y,z) Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have d(x, y) = 0 for distinct values x \neq y.


Examples

Any metric space is a pseudometric space. Pseudometrics arise naturally in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
. Consider the space \mathcal(X) of real-valued functions f : X \to \R together with a special point x_0 \in X. This point then induces a pseudometric on the space of functions, given by d(f,g) = \left, f(x_0) - g(x_0)\ for f, g \in \mathcal(X) A
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
p induces the pseudometric d(x, y) = p(x - y). This is a
convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poi ...
of an
affine function In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generall ...
of x (in particular, a
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
), and therefore convex in x. (Likewise for y.) Conversely, a homogeneous, translation-invariant pseudometric induces a seminorm. Pseudometrics also arise in the theory of hyperbolic
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a ...
s: see Kobayashi metric. Every
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
(\Omega,\mathcal,\mu) can be viewed as a complete pseudometric space by defining d(A,B) := \mu(A \vartriangle B) for all A, B \in \mathcal, where the triangle denotes
symmetric difference In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. Th ...
. If f : X_1 \to X_2 is a function and ''d''2 is a pseudometric on ''X''2, then d_1(x, y) := d_2(f(x), f(y)) gives a pseudometric on ''X''1. If ''d''2 is a metric and ''f'' is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
, then ''d''1 is a metric.


Topology

The is the
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 ...
generated by the open balls B_r(p) = \, which form a basis for the topology. A topological space is said to be a if the space can be given a pseudometric such that the pseudometric topology coincides with the given topology on the space. The difference between pseudometrics and metrics is entirely topological. That is, a pseudometric is a metric if and only if the topology it generates is T0 (that is, distinct points are
topologically distinguishable In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...
). The definitions of
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
s and metric completion for metric spaces carry over to pseudometric spaces unchanged.


Metric identification

The vanishing of the pseudometric induces an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
, called the metric identification, that converts the pseudometric space into a full-fledged
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 setti ...
. This is done by defining x\sim y if d(x,y)=0. Let X^* = X/ be the quotient space of X by this equivalence relation and define \begin d^*:(X/\sim)&\times (X/\sim) \longrightarrow \R_ \\ d^*( &=d(x,y) \end This is well defined because for any x' \in /math> we have that d(x, x') = 0 and so d(x', y) \leq d(x, x') + d(x, y) = d(x, y) and vice versa. Then d^* is a metric on X^* and (X^*,d^*) is a well-defined metric space, called the metric space induced by the pseudometric space (X, d). The metric identification preserves the induced topologies. That is, a subset A \subseteq X is open (or closed) in (X, d) if and only if \pi(A) = /math> is open (or closed) in \left(X^*, d^*\right) and A is saturated. The topological identification is the
Kolmogorov quotient In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing the ...
. An example of this construction is the completion of a metric space by its
Cauchy sequences In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
.


See also

* * * *


Notes


References

* * * * * {{DEFAULTSORT:Pseudometric Space Properties of topological spaces Metric geometry