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,
> Alternatively, if (Y,d) is complete, then Y contains all its limit
> 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).