Inmathematics, apseudometric space is ageneralization of ametric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced byĐuro Kurepa^[1][2] in 1934. In the same way as everynormed space is ametric space, everyseminormed space is a pseudometric space. Because of this analogy, the termsemimetric space (which has a different meaning intopology) is sometimes used as a synonym, especially infunctional analysis.

When a topology is generated using a family of pseudometrics, the space is called agauge space.

A pseudometric space${\displaystyle (X,d)}$ is a set${\displaystyle X}$ together with a non-negativereal-valued function${\displaystyle d:X\times X⟶{\mathbb{R}}_{\ge 0},}$ called apseudometric, 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:${\displaystyle d(x,z)\le d(x,y)+d(y,z)}$

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

Any metric space is a pseudometric space. Pseudometrics arise naturally infunctional 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${\displaystyle f,g\in \mathcal{F}(X)}$

Aseminorm${\displaystyle p}$ induces the pseudometric${\displaystyle d(x,y)=p(x-y)}$. This is aconvex function of anaffine function of${\displaystyle x}$ (in particular, atranslation), 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 ofhyperboliccomplex manifolds: seeKobayashi metric.

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

If${\displaystyle f:X_1\to X_2}$ is a function andd₂ is a pseudometric onX₂, then${\displaystyle d_1(x,y):=d_2(f(x),f(y))}$ gives a pseudometric onX₁. Ifd₂ is a metric andf isinjective, thend₁ is a metric.

Thepseudometric topology is thetopology generated by theopen ballswhich form abasis for the topology.^[3] A topological space is said to be apseudometrizable 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 isT₀ (that is, distinct points aretopologically distinguishable).

The definitions ofCauchy sequences andmetric completion for metric spaces carry over to pseudometric spaces unchanged.^[5]

The vanishing of the pseudometric induces anequivalence relation, called themetric identification, that converts the pseudometric space into a full-fledgedmetric space. This is done by defining${\displaystyle x\sim y}$ if${\displaystyle d(x,y)=0}$. Let${\displaystyle X^{\ast }=X/\sim }$ be thequotient space of${\displaystyle X}$ by this equivalence relation and defineThis is well defined because for any${\displaystyle x^{\prime }\in [x]}$ we have that${\displaystyle d(x,x^{\prime })=0}$ and so${\displaystyle d(x^{\prime },y)\le d(x,x^{\prime })+d(x,y)=d(x,y)}$ and vice versa. Then${\displaystyle d^{\ast }}$ is a metric on${\displaystyle X^{\ast }}$ and${\displaystyle (X^{\ast },d^{\ast })}$ is a well-defined metric space, called themetric 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${\displaystyle \left(X^{\ast },d^{\ast }\right)}$ and${\displaystyle A}$ issaturated. The topological identification is theKolmogorov quotient.

An example of this construction is thecompletion of a metric space by itsCauchy sequences.

• 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

• 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 onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.
• "Example of pseudometric space".PlanetMath.

• Metric geometry
• Properties of topological spaces

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