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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.
In January 2017, Balfanz, along with asimo3089, uploaded Jailbreak, a cops-and-robbers game, to Roblox. On its first day of release, it reached 70,000 concurrent players, a number which Balfanz later said had shocked him. [1] It quickly became one of the most popular games on the platform, and made Balfanz a millionaire. [4] [3]
The Roblox Studio interface as of August 2024. Roblox Studio is the platforms game engine [31] and game development software. [32] [33] The engine, and all games made on Roblox, predominantly uses Luau, [34] a dialect of the Lua 5.1 programming language. [35] Since November 2021, the programming language has been open sourced under the MIT License.
Every metric space is Tychonoff; every pseudometric space is completely regular. Every locally compact regular space is completely regular, and therefore every locally compact Hausdorff space is Tychonoff. In particular, every topological manifold is Tychonoff. Every totally ordered set with the order topology is Tychonoff.
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.}
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.