# Pseudometric space

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

- .
- (
*symmetry*) - (
*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 d
_{2}is a pseudometric on X_{2}, then gives a pseudometric on X_{1}. If d_{2}is a metric and f is injective, then d_{1}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 T_{0} (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, 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 . The topological identification is the Kolmogorov quotient.

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

## Notes

- ↑ "Pseudometric topology".
*PlanetMath*. - ↑ 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 .

- ↑ 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*.