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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.

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

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 .

Psuedometrics 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.

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 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 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 ^* ).

1. ^ 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