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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.
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 ...
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 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.
One common type of approach is claiming to have solved a classical problem that has been proven to be mathematically unsolvable. Common examples of this include the following constructions in Euclidean geometry—using only a compass and straightedge:
Pseudometric may refer to: The metric of a pseudo-Riemannian manifold , a non-degenerate, smooth, symmetric tensor field of arbitrary signature Pseudometric space , a generalization of a metric that does not necessarily distinguish points (and so typically used to study certain non-Hausdorff spaces)
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.
A complex space X is said to be Kobayashi hyperbolic if the Kobayashi pseudometric d X is a metric, meaning that d X (x,y) > 0 for all x ≠ y in X. Informally, this means that there is a genuine bound on the size of discs mapping holomorphically into X.