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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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 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 1, 2013 5:16 PM

On Feb 1, 10:37 am, "Daniel J. Greenhoe" <dgreen...@yahoo.com> 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.

Do you happen to know an example of a metric space which is not
complete? If so, let (X,d) be that metric space, and let Y = X.

If not, do you know an example of a metric space in which some subset
is not closed? In that case, let (W,d) be that metric space, let X be
a non-closed subset of W, and let Y = X. Then (X,d) is an incomplete
metric space, (Y,d) is a closed subspace of (X,d), and (Y,d) is not
complete.

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