Date: Feb 1, 2013 10:14 PM
Author: William Elliot
Subject: closed but not complete

On Fri, 1 Feb 2013, Daniel J. Greenhoe wrote:

A long title in ignarance of what he once learned in composition class.

> 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?

No. Assume K is a closed subset of the complete space (S,d).
Let s be Cauchy sequence into K. Then s is a Cauchy sequence
into S. Hence there's some x in S with s -> x. Since K is closed x in K.

Exercise. Show if S any topological space, K a closed subset of S,
and s a sequence into K that converges into x, then x is in K.

Conclusion. K subset complete S implies (K closed iff K complete).