From: Fred Galvin <galvin@math.ukans.edu>
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.