Paul
Posts:
492
Registered:
7/12/10


Re: looking for example of closed set that is *not* complete in a metric space
Posted:
Feb 3, 2013 6:00 AM


On Sunday, February 3, 2013 9:32:05 AM UTC, Daniel J. Greenhoe wrote: > On Sunday, February 3, 2013 1:24:54 AM UTC+8, quasi wrote: > > > Suppose (X,d) is not complete. Then there must exist a > > > Cauchy sequence in X which does not converge. Let Y be the > > > set of distinct elements of that Cauchy sequence. Then any > > > infinite subset of Y is closed in X but not complete. > > > > Sorry to bother you again. I still don't follow. > > Why is Y closed in (X,d)? >
Many mathematical terms, such as "closed" have many equivalent definitions. An important technique is to pick the definition that makes your task easiest. I would suggest the following definition. Y is closed in X if every limit point of Y is contained in Y. "limit point" can be easily googled if you don't know what this means. Then, so long as you understand what all the terms mean, and are familiar with the techniques of basic mathematical proof, you can show that Y is closed in (X, d).
Paul Epstein

