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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)[reply]

"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)[reply]

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.

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

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

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

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

The article contains this sentence:

"A metric space ${\displaystyle (M,d)}$ is totally bounded if and only if for every real number ${\displaystyle \varepsilon >0}$, there exists a finite collection of open balls of radius ${\displaystyle \varepsilon }$ 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?