
closed but not complete
Posted:
Feb 1, 2013 10:14 PM


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).

