
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

