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Topic: looking for example of closed set that is *not* complete in a metric space
Replies: 26   Last Post: Feb 3, 2013 11:06 AM

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 quasi Posts: 12,012 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

Date Subject Author
2/1/13 Achimota
2/1/13 Paul
2/1/13 Paul
2/1/13 fom
2/1/13 fom
2/2/13 Shmuel (Seymour J.) Metz
2/3/13 fom
2/3/13 Shmuel (Seymour J.) Metz
2/2/13 Achimota
2/2/13 Butch Malahide
2/2/13 quasi
2/2/13 Butch Malahide
2/2/13 Achimota
2/2/13 quasi
2/3/13 Achimota
2/3/13 Paul
2/3/13 Achimota
2/1/13 Butch Malahide
2/1/13 J. Antonio Perez M.
2/1/13 William Hughes
2/2/13 J. Antonio Perez M.
2/1/13 Butch Malahide
2/1/13 William Elliot
2/2/13 Butch Malahide
2/2/13 William Elliot
2/2/13 Butch Malahide
2/2/13 Shmuel (Seymour J.) Metz