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 ]
 Butch Malahide Posts: 894 Registered: 6/29/05
Re: looking for example of closed set that is *not* complete in a
metric space

Posted: Feb 2, 2013 12:32 AM

On Feb 1, 11:10 pm, "Daniel J. Greenhoe" <dgreen...@yahoo.com> wrote:
> On Saturday, February 2, 2013 12:52:55 AM UTC+8, peps...@gmail.com wrote:
> > ...To say that a space is "closed"
> > (as in your statement "closed -> complete") doesn't really mean anything.
> > To make progress replace "closed -> complete" by something more
> > formal and rigorous and precise.

>
> This is certainly good advice and many apologies for my sloppy original posting. Is the following any better?...
>
> Let (X,d) be a metric space.
> Let T be the topology induced by d and
> (X,T) be the resulting topological space.
> Let Y be a subset of X.
> Then
>   (Y,d) is complete ==> Y is closed in (X,d).
> Alternatively,
>   (Y,d) is complete ==> Y is closed in (X,T).
>
> But what about the converse? That is, is this true?
>   Y is closed in (X,d) ?==>? (Y,d) is complete

Is (X,d) complete? If (X,d) is a complete metric space, then every
closed subspace of (X,d) is complete. If (X,d) is not complete, then
it has at least one closed subspace which is not complete, namely,
(X,d) is a closed subspace of itself.

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