Pseudometric space

Wikipedia, the free encyclopedia - Cite This Source

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.

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 $x,y,z\; in\; X$,

1. $,!d(x,x)\; =\; 0$.
2. $,!d(x,y)\; =\; d(y,x)$ (symmetry)
3. $,!d(x,z)\; leq\; d(x,y)\; +\; d(y,z)$ (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 $xne\; y$.

Examples

Pseudometrics arise naturally in functional analysis. Consider the space $mathcal\{F\}(X)$ of real-valued functions $f:Xtomathbb\{R\}$ together with a special point $x\_0in\; X$. This point then induces a pseudometric on the space of functions, given by

$,!d(f,g)\; =\; |f(x\_0)-g(x\_0)|;$

for $f,gin\; mathcal\{F\}(X)$

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

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 $xsim\; y$ if $d(x,y)=0$. Let $X^*=X/sim$ 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 $Asubset\; X$ is open (or closed) in $(X,d)$ if and only if $pi(A)=[A]$ is open (or closed) in $(X^*,d^*)$.

Notes

References

Wikipedia, the free encyclopedia © 2001-2006 Wikipedia contributors (Disclaimer)
This article is licensed under the GNU Free Documentation License.
Last updated on Thursday July 10, 2008 at 12:37:05 PDT (GMT -0700)
View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation