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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 ]
 Butch Malahide Posts: 894 Registered: 6/29/05
Re: closed but not complete
Posted: Feb 2, 2013 5:26 AM

On Feb 2, 2:36 am, William Elliot <ma...@panix.com> wrote:
> On Fri, 1 Feb 2013, Butch Malahide wrote:
> > On Feb 1, 9:14 pm, William Elliot <ma...@panix.com> wrote:
> > > On Fri, 1 Feb 2013, 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?

>
> > > No. Assume K is a closed subset of the complete space (S,d).
>
> > But the original poster did not say that his metric space (X,d) was
> > complete.

>
> Oh, so any closed subset of Q is an example.

*Some* closed subsets of Q are examples.

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