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

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: Uncountable?
Posted: Apr 25, 2014 3:57 AM
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On Fri, 25 Apr 2014, quasi wrote:
> >William Elliot wrote:
> >

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

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

Let qj be the j-th rational and Uj = (qj - 2^-j, qj + 2^-j).
C = { Uj | j in N } covers Q. The measure of \/C <= 2 and
the measure of R is oo. Thus most of R is outside of \/C.

Can it be shown directly without measure theory that R - \/C is uncountable?

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

This problem was inspired by a post in Ask-a-Topologist
asking to show the Micheal Line isn't Lindelof.

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