quasi
Posts:
9,078
Registered:
7/15/05
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Re: looking for example of closed set that is *not* complete in a metric space
Posted:
Feb 2, 2013 12:24 PM
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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
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