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Topic: Uncountable?
Replies: 9   Last Post: Apr 26, 2014 2:13 PM

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quasi

Posts: 10,411
Registered: 7/15/05
Re: Uncountable?
Posted: Apr 25, 2014 3:31 AM
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Virgil wrote:
>William Elliot wrote:
>

>>Let C be a collection of open subsets of R that cover Q.
>>Can S = R - \/C ever be uncountable?

>
>It cannot even be non-empty!
>
>Every open cover of Q as a subset of R covers R.


That's not true.

As a simple example, let

A = (-oo,sqrt(2))

B = (sqrt(2),oo))

Then A \/ B covers Q but doesn't cover R.

If fact, there exist open covers of Q with uncountable complement.

For a construction, see my reply to William Elliot's first thread
on this question.

quasi



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