Pseudometric space

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 $d:X\times X\longrightarrow \mathbb {R} _{\geq 0},$ called a pseudometric, such that for every $x,y,z\in X,$

$d(x,x)=0.$
Symmetry: $d(x,y)=d(y,x)$
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 $x\neq y.$

Examples

Any metric space is a pseudometric space. Pseudometrics arise naturally in functional analysis. Consider the space ${\mathcal {F}}(X)$ of real-valued functions $f:X\to \mathbb {R}$ together with a special point $x_{0}\in X.$ This point then induces a pseudometric on the space of functions, given by $d(f,g)=\left|f(x_{0})-g(x_{0})\right|$ for $f,g\in {\mathcal {F}}(X)$

A seminorm $p$ 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.

Every measure space $(\Omega ,{\mathcal {A}},\mu )$ can be viewed as a complete pseudometric space by defining $d(A,B):=\mu (A\vartriangle B)$ for all $A,B\in {\mathcal {A}},$ 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 pseudometric topology is the topology generated by the open balls $B_{r}(p)=\{x\in X:d(p,x)<r\},$ which form a basis for the topology.^[3] A topological space is said to be a pseudometrizable space^[4] 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.^[5]

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 define ${\begin{aligned}d^{*}:(X/\sim )&\times (X/\sim )\longrightarrow \mathbb {R} _{\geq 0}\\d^{*}([x],[y])&=d(x,y)\end{aligned}}$ This 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 $(X,d)$ .^[6]^[7]

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.

Notes

^ Kurepa, Đuro (1934). "Tableaux ramifiés d'ensembles, espaces pseudodistaciés". C. R. Acad. Sci. Paris. 198 (1934): 1563–1565.
^ Collatz, Lothar (1966). Functional Analysis and Numerical Mathematics. New York, San Francisco, London: Academic Press. p. 51.
^ "Pseudometric topology". PlanetMath.
^ Willard, p. 23
^ Cain, George (Summer 2000). "Chapter 7: Complete pseudometric spaces" (PDF). Archived (PDF) from the original on 7 October 2020. Retrieved 7 October 2020.
^ Howes, Norman R. (1995). Modern Analysis and Topology. New York, NY: Springer. p. 27. ISBN 0-387-97986-7. Retrieved 10 September 2012. 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 $Y$ .
^ Simon, Barry (2015). A comprehensive course in analysis. Providence, Rhode Island: American Mathematical Society. ISBN 978-1470410995.

References

Arkhangel'skii, A.V.; Pontryagin, L.S. (1990). General Topology I: Basic Concepts and Constructions Dimension Theory. Encyclopaedia of Mathematical Sciences. Springer. ISBN 3-540-18178-4.
Steen, Lynn Arthur; Seebach, Arthur (1995) [1970]. Counterexamples in Topology (new ed.). Dover Publications. ISBN 0-486-68735-X.
Willard, Stephen (2004) [1970], General Topology (Dover reprint of 1970 ed.), Addison-Wesley
This article incorporates material from Pseudometric space on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
"Example of pseudometric space". PlanetMath.

[1] Kurepa, Đuro (1934). "Tableaux ramifiés d'ensembles, espaces pseudodistaciés". C. R. Acad. Sci. Paris. 198 (1934): 1563–1565.

[2] Collatz, Lothar (1966). Functional Analysis and Numerical Mathematics. New York, San Francisco, London: Academic Press. p. 51.

[3] "Pseudometric topology". PlanetMath.

[4] Willard, p. 23

[5] Cain, George (Summer 2000). "Chapter 7: Complete pseudometric spaces" (PDF). Archived (PDF) from the original on 7 October 2020. Retrieved 7 October 2020.

[6] Howes, Norman R. (1995). Modern Analysis and Topology. New York, NY: Springer. p. 27. ISBN 0-387-97986-7. Retrieved 10 September 2012. 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 $Y$ .

[7] Simon, Barry (2015). A comprehensive course in analysis. Providence, Rhode Island: American Mathematical Society. ISBN 978-1470410995.

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