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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

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Achimota

Posts: 254
Registered: 4/30/07
Re: looking for example of closed set that is *not* complete in a
metric space

Posted: Feb 2, 2013 12:10 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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

One might guess that it is not true. So would someone happen to know of a counterexample in which the set Y is closed in (X,d), but yet (Y,d) is *not* complete?

References:
1. Kubrusly(2011) Theorem 3.40 page 129:
books.google.com.tw/books?vid=ISBN0817649980&pg=PA129

2. Haaser(1991) 6.10 Proposition page 75:
books.google.com.tw/books?vid=ISBN0486665097&pg=PA75


Many thanks in advance,
Dan






On Saturday, February 2, 2013 12:52:55 AM UTC+8, peps...@gmail.com wrote:
> On Friday, February 1, 2013 4:37:40 PM UTC, 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
>
>
>
> You need to understand that "closed" and "open" don't characterize topologies.
>
> Rather "X is open in Y" describes a relationship between X and Y.
>
> To say that a space is complete or compact or Hausdorff makes a statement about a topological space. 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.
>
>
>
> Paul Epstein




Date Subject Author
2/1/13
Read looking for example of closed set that is *not* complete in a metric space
Achimota
2/1/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
Paul
2/1/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
Paul
2/1/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
fom
2/1/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
fom
2/2/13
Read Re: looking for example of closed set that is *not* complete in a metric space
Shmuel (Seymour J.) Metz
2/3/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
fom
2/3/13
Read Re: looking for example of closed set that is *not* complete in a metric space
Shmuel (Seymour J.) Metz
2/2/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
Achimota
2/2/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
Butch Malahide
2/2/13
Read Re: looking for example of closed set that is *not* complete in a metric space
quasi
2/2/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
Butch Malahide
2/2/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
Achimota
2/2/13
Read Re: looking for example of closed set that is *not* complete in a metric space
quasi
2/3/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
Achimota
2/3/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
Paul
2/3/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
Achimota
2/1/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
Butch Malahide
2/1/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
J. Antonio Perez M.
2/1/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
William Hughes
2/2/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
J. Antonio Perez M.
2/1/13
Read Re: looking for example of closed set that is *not* complete in a
metric space
Butch Malahide
2/1/13
Read closed but not complete
William Elliot
2/2/13
Read Re: closed but not complete
Butch Malahide
2/2/13
Read Re: closed but not complete
William Elliot
2/2/13
Read Re: closed but not complete
Butch Malahide
2/2/13
Read Re: closed but not complete
Shmuel (Seymour J.) Metz

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