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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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Paul

Posts: 432
Registered: 7/12/10
Re: looking for example of closed set that is *not* complete in a
metric space

Posted: Feb 3, 2013 6:00 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Sunday, February 3, 2013 9:32:05 AM UTC, Daniel J. Greenhoe wrote:
> On Sunday, February 3, 2013 1:24:54 AM UTC+8, quasi wrote:
>

> > Suppose (X,d) is not complete. Then there must exist a
>
> > Cauchy sequence in X which does not converge. Let Y be the
>
> > set of distinct elements of that Cauchy sequence. Then any
>
> > infinite subset of Y is closed in X but not complete.
>
>
>
> Sorry to bother you again. I still don't follow.
>
> Why is Y closed in (X,d)?
>


Many mathematical terms, such as "closed" have many equivalent definitions. An important technique is to pick the definition that makes your task easiest.
I would suggest the following definition. Y is closed in X if every limit point of Y is contained in Y. "limit point" can be easily googled if you don't know what this means. Then, so long as you understand what all the terms mean, and are familiar with the techniques of basic mathematical proof, you can show that Y is closed in (X, d).

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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