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Re: looking for example of closed set that is *not* complete in a
metric space
Posted:
Feb 1, 2013 5:16 PM
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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.
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