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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.
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.
The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, since the definition of a topology requires these two sets to be open.
A pseudometric space (,) (for example, a metric space) is called complete and is called a complete pseudometric if any of the following equivalent conditions hold: Every Cauchy prefilter on X {\displaystyle X} converges to at least one point of X . {\displaystyle X.}
The Carathéodory metric is another intrinsic pseudometric on complex manifolds, based on holomorphic maps to the unit disc rather than from the unit disc. The Kobayashi infinitesimal pseudometric is a Finsler pseudometric whose associated distance function is the Kobayashi pseudometric as defined above. [ 18 ]
This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0. But "having distance 0" is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of M .
All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular, although not in general Hausdorff; All compact Hausdorff spaces are normal; In particular, the Stone–Čech compactification of a Tychonoff space is normal Hausdorff;