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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 compact if it is complete and totally bounded. (This definition is written in terms of metric properties and does not make sense for a general topological space, but it is nevertheless topologically invariant since it is equivalent to compactness.) One example of a compact space is the closed interval [0, 1].
The definition of the Hausdorff distance can be derived by a series of natural extensions of the distance function (,) in the underlying metric space M, as follows: [7] Define a distance function between any point x of M and any non-empty set Y of M by (,) = {(,)}.
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 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.
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 T 0 space is a topological space in which every pair of distinct points is topologically distinguishable. ... there is a more direct definition of pseudometric.)