Date: Feb 1, 2013 5:25 PM
Author: J. Antonio Perez M.
Subject: Re: looking for example of closed set that is *not* complete in a<br> metric space

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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> Many thanks in advance,
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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