A theorem on dimension
Morton L. Curtis, G. S. Young · Proceedings of the American Mathematical Society · 1952
Every »-dimensional separable metric space can be homeomorphically imbedded in the closed (2w+l)-cube I2n+1.This, of course, does not characterize «-dimensionality, for spaces of dimension greater than n can be imbedded homeomorphically in I2n+l.A theorem due to Hurewicz1 suggests that the existence of a more general kind of mapping into the n-cube I" may characterize »-dimensionality for compact spaces.For compact spaces this theorem reduces to: If X, Y are compact and separable metric, f:X-*Y is continuous, and dim X = dim Y, then there exists yEY such that