Pseudometric space: Information from Answers.com

In mathematics, a pseudometric 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.

1 Definition
2 Examples
3 Topology
4 Metric identification
5 Notes
6 References

Definition

A pseudometric space $(X, d)$ is a set $X$ together with a non-negative real-valued function $d: X \times X \longrightarrow \mathbb{R}_{\geq 0}$ (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 $x\ne y$ .

Examples

Pseudometrics 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.

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.

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 $x ˜ y$ 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 $(X *, d *)$ .

Notes

^ Pseudometric topology on PlanetMath

References

Arkhangel'skii, A.V.; Pontryagin, L.S. (1990). General Topology I: Basic Concepts and Constructions Dimension Theory. Encyclopaedia of Mathematical Sciences. Springer. ISBN 3-540-18178-4.
Steen, Lynn Arthur; Seebach, Arthur (1995) [1970]. Counterexamples in Topology (new edition ed.). Dover Publications. ISBN 048668735X.
This article incorporates material from Pseudometric space on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Example of pseudometric space on PlanetMath

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

	What is in space? Read answer...
	What does space have in it? Read answer...
	What is out in space? Read answer...

	How do you get to space?
	What do you have to do in space?
	What was on space?

Pseudometric space

Contents