Pseudometric space

In mathematics, a pseudometric or semi-metric space is a generalized metric space in which the distance between two distinct points can be zero. 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

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 $x\neq y$ .

Examples

$d(f,g)=|f(x_{0})-g(x_{0})|$ for $f,g\in {\mathcal {F}}(X)$ $d(x,y)=p(x-y).$ Conversely, a homogeneous, translation invariant pseudometric induces a seminorm.
$d(A,B):=\mu (A\Delta B)$ for all $A,B\in {\mathcal {A}}$ .

Topology

The pseudometric topology is the topology induced by the open balls

$B_{r}(p)=\{x\in X\mid d(p,x) which form a basis for the topology. A topological space is said to be a pseudometrizable topological space 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 (i.e. distinct points are topologically distinguishable).

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 $x\sim y$ if $d(x,y)=0$ . Let $X^{*}=X/{\sim }$ and let

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

An example of this construction is the completion of a metric space by its Cauchy sequences.