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.
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Definition
A pseudometric space (X,d) is a set X together with a nonnegative realvalued function (called a pseudometric) such that, for every ,
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 .
Examples
Pseudometrics arise naturally in functional analysis. Consider the space of realvalued 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.
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).
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