```Date: Feb 1, 2013 1:09 PM
Author: Paul
Subject: Re: looking for example of closed set that is *not* complete in a<br> metric space

On Friday, February 1, 2013 4:52:55 PM UTC, peps...@gmail.com wrote:> 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.> > > > Paul EpsteinTo clarify, you did attempt more precision by saying "closed with respect to d" but you're misusing/misunderstanding the concept of "closed" here, and you need to review your notes.  "closed with respect to d" is not correct.Paul
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