An imbedding theorem for metric spaces

Abstract:

A simple solution to the imbedding problem for the class of separable metric spaces has been known for a long time: (Urysohn’s Imbedding Theorem) A topological space is separable metric if, and only if, it can be imbedded in the topological product of countably many unit intervals. We see that products of the unit interval make an especially informative type of imbedding space since the finite (Lebesgue) dimensional separable metric spaces are those that can be imbedded in a finite product of intervals. The author has recently shown that this result concerning the finite case could be extended to arbitrary metric spaces if we use a topological generalization of the unit interval. This present paper shows that if we use this same generalization of the interval, then we can obtain an analogue to Urysohn’s Imbedding Theorem. Besides presenting the first unified results which simultaneously generalize both separable cases, this paper contains comparisons with existing imbedding theorems and imbedding spaces.