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




Re: Uncountable?
Posted:
Apr 25, 2014 3:57 AM


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 jth 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 AskaTopologist asking to show the Micheal Line isn't Lindelof.



