On Friday, February 1, 2013 4:37:40 PM UTC, 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
You need to understand that "closed" and "open" don't characterize topologies. Rather "X is open in Y" describes a relationship between X and Y. To say that a space is complete or compact or Hausdorff makes a statement about a topological space. To say that a space is "closed" (as in your statement "closed -> complete") doesn't really mean anything. To make progress replace "closed -> complete" by something more formal and rigorous and precise.