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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 ...
The Gromov–Hausdorff space is path-connected, complete, and separable. [5] It is also geodesic, i.e., any two of its points are the endpoints of a minimizing geodesic. [6] [7] In the global sense, the Gromov–Hausdorff space is totally heterogeneous, i.e., its isometry group is trivial, [8] but locally there are many nontrivial isometries.
The Niemytzki plane is an example of a Tychonoff space that is not normal. There are regular Hausdorff spaces that are not completely regular, but such examples are complicated to construct. One of them is the so-called Tychonoff corkscrew , [ 3 ] [ 4 ] which contains two points such that any continuous real-valued function on the space has the ...
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.
Formally, a metric measure space is a metric space equipped with a Borel regular measure such that every ball has positive measure. [21] For example Euclidean spaces of dimension n, and more generally n-dimensional Riemannian manifolds, naturally have the structure of a metric measure space, equipped with the Lebesgue measure.
In mathematics, the concept of a generalised metric is a generalisation of that of a metric, in which the distance is not a real number but taken from an arbitrary ordered field. In general, when we define metric space the distance function is taken to be a real-valued function .
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.