Date: Feb 1, 2013 5:32 PM
Author: William Hughes
Subject: Re: looking for example of closed set that is *not* complete in a<br> metric space
On Feb 1, 11:25 pm, Tonic...@yahoo.com wrote:

> On Friday, February 1, 2013 6:37:40 PM UTC+2, Daniel J. Greenhoe wrote:

> > Let (Y,d) be a subspace of a metric space (X,d).

>

> > If (Y,d) is complete, then Y is closed with respect to d. That is,

>

> > complete==>closed.

>

> > Alternatively, if (Y,d) is complete, then Y contains all its limit

>

> > points.

>

> > Would anyone happen to know of a counterexample for the converse? That

>

> > is, does someone know of any example that demonstrates that

>

> > closed --> complete

>

> > is *not* true? I don't know for sure that it is not true, but I might

>

> > guess that it is not true.

>

> > Many thanks in advance,

>

> > Dan

>

> Perhaps what you want, if I understand you correctly, is within reach in a very familiar space: take the reals R with the usual, euclidean topology (or look at R as the euclidean metric space we all know: it's the same). This is a complete space, yet the CLOSED subset [0,+oo) isn't complete...

>

> Tonio

Why is [0,+oo) not complete?