Pseudometric space

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In mathematics, a pseudometric or semi-metric space[1] 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 is a set 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 for distinct values .

Examples

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

Topology

The pseudometric topology is the topology induced by the open balls

which form a basis for the topology.[2] 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 if . Let and let

Then is a metric on and is a well-defined metric space.[3]

The metric identification preserves the induced topologies. That is, a subset is open (or closed) in if and only if is open (or closed) in . The topological identification is the Kolmogorov quotient.

An example of this construction is the completion of a metric space by its Cauchy sequences.

Notes

  1. Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, ISBN 0-8218-2129-6.
  2. Template:Planetmath reference
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References

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