|
|
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).
>
>
>
> 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.
>
>
>
> Many thanks in advance,
>
> 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
|
|
