Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Topic: looking for example of closed set that is *not* complete in a metric space
Replies: 26   Last Post: Feb 3, 2013 11:06 AM

 Messages: [ Previous | Next ]
 J. Antonio Perez M. Posts: 2,736 Registered: 12/13/04
Re: looking for example of closed set that is *not* complete in a
metric space

Posted: Feb 2, 2013 6:20 AM

On Saturday, February 2, 2013 12:32:40 AM UTC+2, William Hughes wrote:
> 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?

My bad: was thinking of something complete(ly) different. Of course it is.

Date Subject Author
2/1/13 Achimota
2/1/13 Paul
2/1/13 Paul
2/1/13 fom
2/1/13 fom
2/2/13 Shmuel (Seymour J.) Metz
2/3/13 fom
2/3/13 Shmuel (Seymour J.) Metz
2/2/13 Achimota
2/2/13 Butch Malahide
2/2/13 quasi
2/2/13 Butch Malahide
2/2/13 Achimota
2/2/13 quasi
2/3/13 Achimota
2/3/13 Paul
2/3/13 Achimota
2/1/13 Butch Malahide
2/1/13 J. Antonio Perez M.
2/1/13 William Hughes
2/2/13 J. Antonio Perez M.
2/1/13 Butch Malahide
2/1/13 William Elliot
2/2/13 Butch Malahide
2/2/13 William Elliot
2/2/13 Butch Malahide
2/2/13 Shmuel (Seymour J.) Metz