Universal Space

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

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

REFERENCES:

Fréchet, M. "Les dimensions d'un ensemble abstrait." Math. Ann. 68, 145-168, 1910.

Holsztynski, W. " 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. https://arxiv.org/abs/math.GN/0004091.

Urysohn, P. S. "Sur un espace métrique universel." Bull. de Sciences Math. 5, 1-38, 1927.