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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. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space.
A metric space M is bounded if there is an r such that no pair of points in M is more than distance r apart. [b] The least such r is called the diameter of M. The space M is called precompact or totally bounded if for every r > 0 there is a finite cover of M by open balls of radius r. Every totally bounded space is bounded.
A pseudometric is a generalization of a metric which does not satisfy the condition that (,) = only when =. A locally convex space is pseudometrizable, meaning that its topology arises from a pseudometric, if and only if it has a countable family of seminorms.
A pseudometric (resp. metric) is induced by a seminorm (resp. norm) on a vector space if and only if is translation invariant and absolutely homogeneous, which means that for all scalars and all ,, in which case the function defined by ():= (,) is a seminorm (resp. norm) and the pseudometric (resp. metric) induced by is equal to .
A Baire space is a topological space in which every countable intersection of open dense sets is dense in . See the corresponding article for a list of equivalent characterizations, as some are more useful than others depending on the application. (BCT1) Every complete pseudometric space is a Baire space.
Pseudometric space A pseudometric space (M, d) is a set M equipped with a real-valued function : satisfying all the conditions of a metric space, except possibly the identity of indiscernibles. That is, points in a pseudometric space may be "infinitely close" without being identical. The function d is a pseudometric on M. Every metric is a ...
Almost every topological space studied in mathematical analysis is Tychonoff, or at least completely regular. For example, the real line is Tychonoff under the standard Euclidean topology. Other examples include: Every metric space is Tychonoff; every pseudometric space is completely regular.
A separable measure space has a natural pseudometric that renders it separable as a pseudometric space. The distance between two sets is defined as the measure of the symmetric difference of the two sets. The symmetric difference of two distinct sets can have measure zero; hence the pseudometric as defined above need not to be a true metric.