Pseudometric space

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In mathematics, a pseudometric space is a generalization of a metric space, where the requirement of indistinguishability is removed. A pseudometric space is a special case of a hemimetric space, on which the requirement of symmetry is imposed. When a topology is generated using a family of pseudometrics, the space is called a gauge space.

1 Definition
2 Examples
3 Topology
4 Metric identification
5 References

[edit] Definition

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

$\,\!d(x,x) = 0$ .
$\,\!d(x,y) = d(y,x)$ (symmetry)
$\,\!d(x,z) \leq d(x,y) + d(y,z)$ (subadditivity/triangle inequality)

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

[edit] Examples

Psuedometrics arise naturally in functional analysis. Consider the space $\mathcal{F}(X)$ of real-valued functions $f:X\to\mathbb{R}$ together with a special point $x_0\in X$ . This point then induces a pseudometric on the space of functions, given by

$\,\!d(f,g) = |f(x_0)-g(x_0)|\;$

for $f,g\in \mathcal{F}(X)$

For vector spaces V, a seminorm p induces a pseudometric on V, as

$\,\!d(x,y)=p(x-y)$

Conversely, a homogeneous, translation invariant pseudometric induces a seminorm.

[edit] Topology

The pseudometric topology is the topology induced by the open balls

$B_r(p)=\{ x\in X\mid d(p,x)<r \}$ ,

which form a basis for the topology^[1]. 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.

[edit] 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 $X * = X / ˜$ and let

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

Then $d *$ is a metric on $X *$ and $(X *, d *)$ is a well-defined metric space.

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 $π(A) = [A]$ is open (or closed) in $(X *, d *)$ .

[edit] References

^ Pseudometric topology on PlanetMath

L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, (1970) Holt, Rinehart and Winston, Inc..
A.V. Arkhangelskii, L.S.Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4
This article incorporates material from Pseudometric space on PlanetMath, which is licensed under the GFDL.
Example of pseudometric space on PlanetMath