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

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.


>>>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).