Pseudometric space

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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.

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

[edit] Definition

A pseudometric space (X,d) is a set X together with a non-negative real-valued function (called a pseudometric) such that, for every ,

1. .
2. (symmetry)
3. (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 .

[edit] Examples

Pseudometrics arise naturally in functional analysis. Consider the space of real-valued functions together with a special point . This point then induces a pseudometric on the space of functions, given by

for

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

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

[edit] Topology

The pseudometric topology is the topology induced by the open balls

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).

[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˜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 is open (or closed) in (X,d) if and only if π(A) = [A] is open (or closed) in (X ^* ,d ^* ).

[edit] Notes

1. ^ Pseudometric topology on PlanetMath

[edit] 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 GFDL.
• Example of pseudometric space on PlanetMath