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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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 J. Antonio Perez M. Posts: 2,736 Registered: 12/13/04
Re: looking for example of closed set that is *not* complete in a
metric space

Posted: Feb 1, 2013 5:25 PM

On Friday, February 1, 2013 6:37:40 PM UTC+2, Daniel J. Greenhoe wrote:
> Let (Y,d) be a subspace of a metric space (X,d).
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> If (Y,d) is complete, then Y is closed with respect to d. That is,
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> complete==>closed.
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> Alternatively, if (Y,d) is complete, then Y contains all its limit
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> points.
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> Would anyone happen to know of a counterexample for the converse? That
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> is, does someone know of any example that demonstrates that
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> closed --> complete
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> is *not* true? I don't know for sure that it is not true, but I might
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> guess that it is not true.
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> Dan

Perhaps what you want, if I understand you correctly, is within reach in a very familiar space: take the reals R with the usual, euclidean topology (or look at R as the euclidean metric space we all know: it's the same). This is a complete space, yet the CLOSED subset [0,+oo) isn't complete...

Tonio

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