From: Fred Galvin Subject: Re: Separable metric space Date: Fri, 9 Feb 2001 20:01:27 -0600 Newsgroups: sci.math Summary: Embedding metric spaces into Euclidean spaces On Fri, 9 Feb 2001 azmi_tamid@my-deja.com wrote: > If X is a separable metric space then can we find topological embedding > of X to R^d for some d ? Not if by "some d" you mean "some natural number d". Hilbert space is a separable metric space; being infinite-dimensional, it is not embeddable in any R^d with d finite; it is homeomorphic to R^omega. Every separable metric space is embeddable in Hilbert space. Every finite-dimensional separable metric space is embeddable in some R^d; in fact, every m-dimensional separable metric space is embeddable in R^{2m+1}. At least, that's how I remember it; hope I've got it right. See the book _Dimension Theory_ by Hurewicz and Wallman. -- It takes steel balls to play pinball.