# Pseudometric space

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 ,

- .
- (
*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 vector spaces , a seminorm induces a pseudometric on , as

- 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

- 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.^{[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 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.^{[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

- ↑ Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, ISBN 0-8218-2129-6.
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## References

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*This article incorporates material from Pseudometric space on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*- Template:Planetmath reference