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.

>

>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