Pseudometric space explained

In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa^{[1] [2]} in 1934. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space. Because of this analogy, the term semimetric space (which has a different meaning in topology) is sometimes used as a synonym, especially in functional analysis.

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 x X \longrightarrow \R _{\geq} , called a , such that for every x, y, z \in X, d(x,x) = 0 .

1. Symmetry:

d(x,y) = d(y,x)

1. Subadditivity/Triangle inequality:

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 ≠ y.

Examples

Any metric space is a pseudometric space. Pseudometrics arise naturally in functional analysis. Consider the space

l{F}(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 $$d(f,g)\; =\; \backslash left|f(x\_0)\; -\; g(x\_0)\backslash right|$$ for f, g \in l{F}(X) p induces the pseudometric d(x, y) = p(x - y) . This is a convex function of an affine function of x (in particular, a translation), 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 manifolds: see Kobayashi metric.

(\Omega, l{A},\mu) can be viewed as a complete pseudometric space by defining $$d(A,B)\; :=\; \backslash mu(A\; \backslash vartriangle\; B)$$ for all A, B \in l{A}, where the triangle denotes symmetric difference.

If

f : X _{1} \to X _{2} is a function and d₂ is a pseudometric on X₂, then d _{1(x,} y) : = d _{2(f(x),} f(y)) gives a pseudometric on X₁. If d₂ is a metric and f is injective, then d₁ is a metric.

Topology

The is the topology generated by the open balls$$B\_r(p)\; =\; \backslash ,$$which form a basis for the topology. A topological space is said to be a ^[3] 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).

The definitions of Cauchy sequences and metric completion for metric spaces carry over to pseudometric spaces unchanged.^[4]

Metric identification

The vanishing of the pseudometric induces an equivalence relation, called the metric identification, that converts the pseudometric space into a full-fledged metric space. This is done by defining

x\sim y if d(x,y) =0 . Let X ^{*} = X/{\sim} be the quotient space of X by this equivalence relation and defineThis is well defined because for any x' \in [x] 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) .^{[5] [6]}

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) = [A] is open (or closed) in \left(X ^{*,} d ^{*\right)} and A is saturated. The topological identification is the Kolmogorov quotient.

An example of this construction is the completion of a metric space by its Cauchy sequences.

References

• Book: Arkhangel'skii, A.V. . General Topology I: Basic Concepts and Constructions Dimension Theory . Alexander Arhangelskii. Pontryagin, L.S. . Lev Pontryagin. 1990 . 3-540-18178-4 . . Encyclopaedia of Mathematical Sciences.
• Book: Steen, Lynn Arthur . Counterexamples in Topology . Lynn Arthur Steen. J. Arthur Seebach Jr.. Seebach, Arthur . 1995 . 1970 . 0-486-68735-X . . new .

Notes and References

1. Kurepa. Đuro. 1934. Tableaux ramifiés d'ensembles, espaces pseudodistaciés. C. R. Acad. Sci. Paris. 198 (1934). 1563–1565.
2. Book: Collatz, Lothar. Functional Analysis and Numerical Mathematics. Academic Press. 1966. New York, San Francisco, London. 51. English.
3. Willard, p. 23
4. Web site: Cain. George. Summer 2000. Chapter 7: Complete pseudometric spaces. live. https://archive.today/20201007070509/http://people.math.gatech.edu/~cain/summer00/ch7.pdf. 7 October 2020. 7 October 2020.
5. Book: Howes, Norman R.. Modern Analysis and Topology. 1995. Springer. New York, NY. 0-387-97986-7. 10 September 2012. 27. Let (X,d) be a pseudo-metric space and define an equivalence relation \sim in X by x \sim y if d(x,y) =0 . Let Y be the quotient space X/\sim and p : X\to Y the canonical projection that maps each point of X onto the equivalence class that contains it. Define the metric \rho in Y by \rho(a,b) = d(p ^{-1} (a),p ^{-1} (b)) for each pair a,b \in Y . It is easily shown that \rho is indeed a metric and \rho defines the quotient topology on Y ..
6. Book: Simon, Barry. A comprehensive course in analysis. American Mathematical Society. 2015. 978-1470410995. Providence, Rhode Island.