Pseudometric space#Topology

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

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

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 $x\; \backslash neq\; y.$

Any metric space is a pseudometric space.

Pseudometrics arise naturally in functional analysis. Consider the space $\backslash mathcal\{F\}(X)$ of real-valued functions $f\; :\; X\; \backslash to\; \backslash R$ together with a special point $x\_0\; \backslash in\; X.$ 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 $f,\; g\; \backslash in\; \backslash mathcal\{F\}(X)$

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

If $f\; :\; X\_1\; \backslash to\; X\_2$ 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.

The {{visible anchor|pseudometric topology}} is the topology generated by the open balls

which form a basis for the topology.{{planetmath reference|urlname=PseudometricTopology|title=Pseudometric topology}} A topological space is said to be a {{visible anchor|pseudometrizable space}}Willard, p. 23 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.{{Cite web|last=Cain|first=George|date=Summer 2000|title=Chapter 7: Complete pseudometric spaces|url=http://people.math.gatech.edu/~cain/summer00/ch7.pdf|url-status=live|archive-url=https://archive.today/20201007070509/http://people.math.gatech.edu/~cain/summer00/ch7.pdf|archive-date=7 October 2020|access-date=7 October 2020}}

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 $x\backslash sim\; y$ if $d(x,y)=0$. Let $X^*\; =\; X/\{\backslash sim\}$ be the quotient space of $X$ by this equivalence relation and define

$$\backslash begin\{align\}$$

d^*:(X/\sim)&\times (X/\sim) \longrightarrow \R_{\geq 0} \\

d^*([x],[y])&=d(x,y)

\end{align}

This is well defined because for any $x\text{'}\; \backslash in\; [x]$ we have that $d(x,\; x\text{'})\; =\; 0$ and so $d(x\text{'},\; y)\; \backslash leq\; d(x,\; x\text{'})\; +\; d(x,\; y)\; =\; d(x,\; y)$ 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 $(X,\; d)$.{{cite book|last=Howes|first=Norman R.|title=Modern Analysis and Topology|year=1995|publisher=Springer|location=New York, NY|isbn=0-387-97986-7|url=https://www.springer.com/mathematics/analysis/book/978-0-387-97986-1|access-date=10 September 2012|page=27|quote=Let $(X,d)$ be a pseudo-metric space and define an equivalence relation $\backslash sim$ in $X$ by $x\; \backslash sim\; y$ if $d(x,y)=0$. Let $Y$ be the quotient space $X/\backslash sim$ and $p\; :\; X\backslash to\; Y$ the canonical projection that maps each point of $X$ onto the equivalence class that contains it. Define the metric $\backslash rho$ in $Y$ by $\backslash rho(a,b)\; =\; d(p^\{-1\}(a),p^\{-1\}(b))$ for each pair $a,b\; \backslash in\; Y$. It is easily shown that $\backslash rho$ is indeed a metric and $\backslash rho$ defines the quotient topology on $Y$.}}{{cite book|title=A comprehensive course in analysis|last=Simon|first=Barry|publisher=American Mathematical Society|year=2015|isbn=978-1470410995|location=Providence, Rhode Island}}

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

• {{annotated link|Generalised metric}}
• {{annotated link|Metric signature}}
• {{annotated link|Metric space}}
• {{annotated link|Metrizable topological vector space}}

{{reflist}}

• {{cite book | title=General Topology I: Basic Concepts and Constructions Dimension Theory | last=Arkhangel'skii | first=A.V. |author1link = Alexander Arhangelskii|author2=Pontryagin, L.S. |author2link = Lev Pontryagin| year=1990 | isbn=3-540-18178-4 | publisher=Springer | series=Encyclopaedia of Mathematical Sciences}}
• {{cite book | title=Counterexamples in Topology | last=Steen | first=Lynn Arthur |author1link = Lynn Arthur Steen|author2link = J. Arthur Seebach Jr.|author2=Seebach, Arthur | year=1995 | orig-year=1970 | isbn=0-486-68735-X | publisher=Dover Publications | edition=new }}
• {{Citation | last=Willard | first=Stephen | title=General Topology | orig-year=1970 | publisher=Addison-Wesley | edition=Dover reprint of 1970 | year=2004}}
• {{PlanetMath attribution|id=6273|title=Pseudometric space}}
• {{planetmath reference|urlname=ExampleOfPseudometricSpace|title=Example of pseudometric space}}

{{Metric spaces}}

{{DEFAULTSORT:Pseudometric Space}}

Category:Metric geometry

Category:Properties of topological spaces