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 ${\displaystyle (X,d)}$ is a set ${\displaystyle X}$ together with a non-negative real-valued function ${\displaystyle d:X\times X\longrightarrow \mathbb {R} _{\geq 0},}$ called a pseudometric, such that for every ${\displaystyle x,y,z\in X,}$

1. ${\displaystyle d(x,x)=0.}$
2. Symmetry: ${\displaystyle d(x,y)=d(y,x)}$
3. Subadditivity/Triangle inequality: ${\displaystyle d(x,z)\leq d(x,y)+d(y,z)}$

Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have ${\displaystyle d(x,y)=0}$ for distinct values ${\displaystyle x\neq y.}$

Examples

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

A seminorm ${\displaystyle p}$ induces the pseudometric ${\displaystyle d(x,y)=p(x-y)}$. This is a convex function of an affine function of ${\displaystyle x}$ (in particular, a translation), and therefore convex in ${\displaystyle x}$. (Likewise for ${\displaystyle y}$.)

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

If ${\displaystyle f:X_{1}\to X_{2}}$ is a function and d₂ is a pseudometric on X₂, then ${\displaystyle d_{1}(x,y):=d_{2}(f(x),f(y))}$ 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 $${\displaystyle B_{r}(p)=\{x\in X:d(p,x)<r\},}$$ 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 ${\displaystyle x\sim y}$ if ${\displaystyle d(x,y)=0}$. Let ${\displaystyle X^{*}=X/{\sim }}$ be the quotient space of ${\displaystyle X}$ by this equivalence relation and define $${\displaystyle {\begin{aligned}d^{*}:(X/\sim )&\times (X/\sim )\longrightarrow \mathbb {R} _{\geq 0}\\d^{*}([x],[y])&=d(x,y)\end{aligned}}}$$ This is well defined because for any ${\displaystyle x'\in [x]}$ we have that ${\displaystyle d(x,x')=0}$ and so ${\displaystyle d(x',y)\leq d(x,x')+d(x,y)=d(x,y)}$ and vice versa. Then ${\displaystyle d^{*}}$ is a metric on ${\displaystyle X^{*}}$ and ${\displaystyle (X^{*},d^{*})}$ is a well-defined metric space, called the metric space induced by the pseudometric space ${\displaystyle (X,d)}$.[6][7]

The metric identification preserves the induced topologies. That is, a subset ${\displaystyle A\subseteq X}$ is open (or closed) in ${\displaystyle (X,d)}$ if and only if ${\displaystyle \pi (A)=[A]}$ is open (or closed) in ${\displaystyle \left(X^{*},d^{*}\right)}$ and ${\displaystyle A}$ 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.

See also

• Generalised metric – Metric geometry
• Metric signature – Number of positive, negative and zero eigenvalues of a metric tensor
• Metric space – Mathematical space with a notion of distance
• Metrizable topological vector space – A topological vector space whose topology can be defined by a metric

Notes

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 ${\displaystyle (X,d)}$ be a pseudo-metric space and define an equivalence relation ${\displaystyle \sim }$ in ${\displaystyle X}$ by ${\displaystyle x\sim y}$ if ${\displaystyle d(x,y)=0}$. Let ${\displaystyle Y}$ be the quotient space ${\displaystyle X/\sim }$ and ${\displaystyle p:X\to Y}$ the canonical projection that maps each point of ${\displaystyle X}$ onto the equivalence class that contains it. Define the metric ${\displaystyle \rho }$ in ${\displaystyle Y}$ by ${\displaystyle \rho (a,b)=d(p^{-1}(a),p^{-1}(b))}$ for each pair ${\displaystyle a,b\in Y}$. It is easily shown that ${\displaystyle \rho }$ is indeed a metric and ${\displaystyle \rho }$ defines the quotient topology on ${\displaystyle Y}$.
7. 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.