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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)
On the set of all subsets of M, d H yields an extended pseudometric. On the set F(M) of all non-empty compact subsets of M, d H is a metric. If M is complete, then so is F(M). [6] If M is compact, then so is F(M). The topology of F(M) depends only on the topology of M, not on the metric d.
The Roblox Studio interface as of August 2024. Roblox Studio is the platforms game engine [26] and game development software. [27] [28] The engine and all games made on Roblox predominantly uses Luau, [29] a dialect of the Lua 5.1 programming language. [30] Since November 2021, the programming language has been open sourced under the MIT License.
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 .
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Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge spaces. Indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff. [5]