Pseudometric space

In mathematics, a pseudometric space is a generalization of a 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

  • 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 , a seminorm induces a pseudometric on , as
Conversely, a homogeneous, translation-invariant pseudometric induces a seminorm.
  • Pseudometrics also arise in the theory of hyperbolic complex manifolds: see Kobayashi metric.
  • Every measure space can be viewed as a complete pseudometric space by defining
for all , where the triangle denotes symmetric difference.
  • If is a function and d2 is a pseudometric on X2, then gives a pseudometric on X1. If d2 is a metric and f is injective, then d1 is a metric.

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).

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 be the quotient space of X by this equivalence relation and define

Then is a metric on and is a well-defined metric space, called the metric space induced by the pseudometric space .[2][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 and A is saturated. 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. "Pseudometric topology". PlanetMath.
  2. Howes, Norman R. (1995). Modern Analysis and Topology. New York, NY: Springer. p. 27. ISBN 0-387-97986-7. Retrieved 10 September 2012. Let be a pseudo-metric space and define an equivalence relation in by if . Let be the quotient space and the canonical projection that maps each point of onto the equivalence class that contains it. Define the metric in by for each pair . It is easily shown that is indeed a metric and defines the quotient topology on .
  3. Simon, Barry (2015). A comprehensive course in analysis. Providence, Rhode Island: American Mathematical Society. ISBN 1470410990.

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 ed.). Dover Publications. ISBN 0-486-68735-X.
  • This article incorporates material from Pseudometric space on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
  • "Example of pseudometric space". PlanetMath.
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