Question

what is pseudo metric​

Answer 1

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.

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Answer 2

In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. In the same way as every normed spaceis 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.

DefinitionEdit

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

{\displaystyle d(x,x)=0}.{\displaystyle d(x,y)=d(y,x)} (symmetry){\displaystyle d(x,z)\leqslant d(x,y)+d(y,z)}(subadditivity/triangle inequality)

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}.

ExamplesEdit

Pseudometrics arise naturally in functional analysis. Consider the space {\displaystyle {\mathcal {F}}(X)} of real-valued functions {\displaystyle f\colon 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)=|f(x_{0})-g(x_{0})|}for {\displaystyle f,g\in {\mathcal {F}}(X)}For vector spaces {\displaystyle V}, a seminorm {\displaystyle p} induces a pseudometric on {\displaystyle V}, as{\displaystyle d(x,y)=p(x-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}\rightarrow 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.
Metric identificationEdit

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 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}}}

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)}.[2][3]

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