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. Pseudometric spaces were introduced by Đuro Kurepa in 1934. 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 formula_1  is a set formula_2 together with a non-negative real-valued function formula_3 called a , such that for every formula_4 Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have formula_8 for distinct values formula_9 Examples. Any metric space is a pseudometric space. Pseudometrics arise naturally in functional analysis. Consider the space formula_10 of real-valued functions formula_11 together with a special point formula_12 This point then induces a pseudometric on the space of functions, given by formula_13 for formula_14 For vector spaces formula_15 a seminorm formula_16 induces a pseudometric on formula_17 , as formula_18 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 formula_19 can be viewed as a complete pseudometric space by defining formula_20 for all formula_21 where the triangle denotes symmetric difference. If formula_22 is a function and "d" is a pseudometric on "X", then formula_23 gives a pseudometric on "X". If "d" is a metric and "f" is injective, then "d" is a metric. Topology. The is the topology generated by the open balls formula_24 which form a basis for the topology. A  topological space is said to be a 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 (that is, distinct points are topologically distinguishable). The definitions of Cauchy sequences and metric completion for metric spaces carry over to pseudometric spaces unchanged. 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 formula_25 if formula_26. Let formula_27 be the quotient space of formula_2 by this equivalence relation and define formula_29 This is well defined because for any formula_30 we have that formula_31 and so formula_32 and vice versa. Then formula_33 is a metric on formula_34 and formula_35 is a well-defined metric space, called the metric space induced by the pseudometric space formula_36. The metric identification preserves the induced topologies. That is, a subset formula_37 is open (or closed) in formula_36 if and only if formula_39 is open (or closed) in formula_40 and  formula_41 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.