In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero.
Pseudometric spaces were introduced by Đuro Kurepa[1][2] in 1934.
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.
together with a non-negative real-valued function
Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have
for distinct values
Pseudometrics arise naturally in functional analysis.
of real-valued functions
This point then induces a pseudometric on the space of functions, given by
(in particular, a translation), and therefore convex in
Conversely, a homogeneous, translation-invariant pseudometric induces a seminorm.
Pseudometrics also arise in the theory of hyperbolic complex manifolds: see Kobayashi metric.
can be viewed as a complete pseudometric space by defining
where the triangle denotes symmetric difference.
The pseudometric topology is the topology generated by the open balls
which form a basis for the topology.
[3] A topological space is said to be a pseudometrizable space[4] 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 (that is, distinct points are topologically distinguishable).
The definitions of Cauchy sequences and metric completion for metric spaces carry over to pseudometric spaces unchanged.
[5] The vanishing of the pseudometric induces an equivalence relation, called the metric identification, that converts the pseudometric space into a full-fledged metric space.
by this equivalence relation and define
and vice versa.
is a well-defined metric space, called the metric space induced by the pseudometric space
[6][7] The metric identification preserves the induced topologies.
is open (or closed) in
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.