quasi
Posts:
12,046
Registered:
7/15/05


Re: looking for example of closed set that is *not* complete in a metric space
Posted:
Feb 2, 2013 12:24 PM


Daniel J. Greenhoe wrote: >Butch Malahide wrote: >>quasi wrote >>>Butch Malahide wrote >>>> >>>>If (X,d) is not complete, then it has at least one closed >>>>subspace which is not complete, namely, (X,d) is a closed >>>>subspace of itself. > >Understood. > >>>Moreover, if (X,d) is not complete, it has uncountably many >>>subsets which are closed but not complete. >> >> Oh, right. At least 2^{aleph_0} of them. > >Not understood. Can someone help me understand this one?
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. Since Y is countably infinite, the cardinality of the set of infinite subsets of Y is 2^(aleph_0).
quasi

