Talk:Totally bounded space

Precompact
At http://planetmath.org/encyclopedia/PrecompactSet.html they define "precompact" as a synonym for "relatively compact". This is of course differnt in the non-metric version. Maybe precompact should be a disambiguation not a redirect? A Geek Tragedy 11:04, 19 June 2006 (UTC)

"In a locally convex space endowed with the weak topology the precompact sets are exactly the bounded sets." As a result of Goldstine's Theorem, the closed unit ball in a banach space is weakly compact if and only if the space is reflexive. Therefore, here, precompact doesn't mean relatively compact. Bounded sets and totally bounded sets in a locally convex space should also be defined. Pz0 (talk) 20:01, 22 June 2009 (UTC)

e-net
imho the definition of totally bounded in terms of an ε-net is more common, and also easier to read than the definition presented here. I don't have any web sources to back this up tho.

definition e-net
Let $$(X,d)$$ be a metric space and let $$U \subseteq X$$. A set $$A \subseteq X $$ is an $$\epsilon$$-net for $$U$$ if for every $$ x \in U$$ there is a $$y \in A$$ such that $$d(x,y) < \epsilon $$.

A set $$V$$ is totally bounded if for every $$\epsilon > 0$$ there exists a finite $$\epsilon$$-net of $$V$$.

See "Introductory Real Analysis" by A. N. Kolmogorov and S. V. Fomin.

149.171.6.248 (talk) 00:55, 23 May 2008 (UTC)

Ridiculous statement
The article contains this sentence:

"A metric space $$ (M,d) $$ is totally bounded if and only if for every real number $$\varepsilon > 0$$, there exists a finite collection of open balls of radius $$\varepsilon$$ whose centers lie in M and whose union contains $M$."

But there has been no mention of any space other than M, so it makes no sense to say "whose centers lie in M".

Where else could they be?