Universal Space

A topological space that contains a homeomorphic image of every topological space of a certain class.

A metric space U is said to be universal for a family of metric spaces M if any space from M is isometrically embeddable in U. Fréchet (1910) proved that l^infty, the space of all bounded sequences of real numbers endowed with a supremum norm, is a universal space for the family M of all separable metric spaces. Holsztynski (1978) proved that there exists a metric d on R, inducing the usual topology, such that every finite metric space embeds in (R,d) (Ovchinnikov 2000).

See also

Metric Space

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Fréchet, M. "Les dimensions d'un ensemble abstrait." Math. Ann. 68, 145-168, 1910.Holsztynski, W. "R^n as a Universal Metric Space." Not. Amer. Math. Soc. 25, A-367, 1978.Ovchinnikov, S. "Universal Metric Spaces According to W. Holsztynski." 13 Apr 2000., P. S. "Sur un espace métrique universel." Bull. de Sciences Math. 5, 1-38, 1927.

Referenced on Wolfram|Alpha

Universal Space

Cite this as:

Weisstein, Eric W. "Universal Space." From MathWorld--A Wolfram Web Resource.

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