Editing Pseudometric space (section)

==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 <math>x\sim y</math> if <math>d(x,y)=0</math>. Let <math>X^* = X/{\sim}</math> be the [[Quotient space (topology)|quotient space]] of <math>X</math> by this equivalence relation and define
<math display=block>\begin{align}
  d^*:(X/\sim)&\times (X/\sim) \longrightarrow \R_{\geq 0} \\
  d^*([x],[y])&=d(x,y)
\end{align}</math>
This is well defined because for any <math>x' \in [x]</math> we have that <math>d(x, x') = 0</math> and so <math>d(x', y) \leq d(x, x') + d(x, y) = d(x, y)</math> and vice versa. Then <math>d^*</math> is a metric on <math>X^*</math> and <math>(X^*,d^*)</math> is a well-defined metric space, called the '''metric space induced by the pseudometric space''' <math>(X, d)</math>.<ref>{{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 <math>(X,d)</math> be a pseudo-metric space and define an equivalence relation <math>\sim</math> in <math>X</math> by <math>x \sim y</math> if <math>d(x,y)=0</math>. Let <math>Y</math> be the quotient space <math>X/\sim</math> and <math>p : X\to Y</math> the canonical projection that maps each point of <math>X</math> onto the equivalence class that contains it. Define the metric <math>\rho</math> in <math>Y</math> by <math>\rho(a,b) = d(p^{-1}(a),p^{-1}(b))</math> for each pair <math>a,b \in Y</math>. It is easily shown that <math>\rho</math> is indeed a metric and <math>\rho</math> defines the quotient topology on <math>Y</math>.}}</ref><ref>{{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}}</ref>

The metric identification preserves the induced topologies. That is, a subset <math>A \subseteq X</math> is open (or closed) in <math>(X, d)</math> if and only if <math>\pi(A) = [A]</math> is open (or closed) in <math>\left(X^*, d^*\right)</math> and <math>A</math> is [[Saturated set|saturated]]. The topological identification is the [[Kolmogorov quotient]].

An example of this construction is the [[Complete metric space#Completion|completion of a metric space]] by its [[Cauchy sequences]].