From: Fred Galvin <galvin@math.ukans.edu>
Subject: Re: Covering with totally- bounded sets
Date: Sat, 2 Sep 2000 21:44:04 -0500
Newsgroups: sci.math
Summary: [missing]

On 2 Sep 2000, Jose wrote:

> Let (X,d) be a metric space / there exists a countable subset D of X
> such that its adherence is X. Question:
> There exist Borel sets C_n n=1,2,... such that:
> i) X is the union of C_n n=1,2,...
> ii) Each C_n is totally-bounded (for every E>0 there exist a finite
> subset {x_1,...,x_p} of C_n / C_n is included in the union of the
> balls with center x_i and radious E i=1,...p)?????
> If this is false, is it true when (X,d) is complete????

In an infinite-dimensional Hilbert space, each totally bounded set is
nowhere dense. By the Baire category theorem, countably many such sets
cannot cover the space.

-- 
"Any clod can have the facts, but having opinions is an art."--McCabe