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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 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)
From a categorical point of view, the extended pseudometric spaces and the extended pseudoquasimetric spaces, along with their corresponding nonexpansive maps, are the best behaved of the metric space categories. One can take arbitrary products and coproducts and form quotient objects within the given category.
An additive topological group is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators.. A topology on a real or complex vector space is called a vector topology or a TVS topology if it makes the operations of vector addition and scalar multiplication continuous (that is, if it makes into a topological vector space).
The Kobayashi infinitesimal pseudometric is a Finsler pseudometric whose associated distance function is the Kobayashi pseudometric as defined above. [18] The Kobayashi–Eisenman pseudo-volume form is an intrinsic measure on a complex n -fold, based on holomorphic maps from the n -dimensional polydisc to X .
A set X is a harmonious set if, for every character χ on the additive closure of X and every ε > 0, there exists a continuous character on the whole space that ε-approximates χ. Then a relatively dense set X is a Meyer set if and only if it is harmonious. [1]
(BCT1) Every complete pseudometric space is a Baire space. [9] [10] In particular, every completely metrizable topological space is a Baire space. (BCT2) Every locally compact regular space is a Baire space. [9] [11] In particular, every locally compact Hausdorff space is a Baire space. BCT1 shows that the following are Baire spaces:
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.