Date: Feb 2, 2013 6:20 AM
Author: J. Antonio Perez M.
Subject: Re: looking for example of closed set that is *not* complete in a<br> metric space

On Saturday, February 2, 2013 12:32:40 AM UTC+2, William Hughes wrote:
> On Feb 1, 11:25 pm, Tonic...@yahoo.com wrote:
>

> > On Friday, February 1, 2013 6:37:40 PM UTC+2, Daniel J. Greenhoe wrote:
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> > > Let (Y,d) be a subspace of a metric space (X,d).
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> >
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> > > If (Y,d) is complete, then Y is closed with respect to d. That is,
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> >
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> > >   complete==>closed.
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> >
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> > > Alternatively, if (Y,d) is complete, then Y contains all its limit
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> >
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> > > points.
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> >
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> > > Would anyone happen to know of a counterexample for the converse? That
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> >
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> > > is, does someone know of any example that demonstrates that
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> >
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> > >    closed --> complete
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> >
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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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> >
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> > > guess that it is not true.
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> >
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> > > Many thanks in advance,
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> >
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> > > Dan
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> >
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> > 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...
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> >
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> > Tonio
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>
> Why is [0,+oo) not complete?



My bad: was thinking of something complete(ly) different. Of course it is.