
Re: Uncountable?
Posted:
Apr 25, 2014 4:35 AM


In article <Pine.NEB.4.64.1404250003140.11979@panix2.panix.com>, William Elliot <marsh@panix.com> wrote:
> Let C be a collection of open subsets of R that cover Q. > Can S = R  \/C ever be uncountable?
Yes, since the cover can be as "small" as you want (by that I mean its measure). Number the elements of Q somehow: q1, q2, q3 ... qn... (this can be done since Q is countable). Cover q1 by an interval of size 1, q2 by an interval of size 1/2, and so on; the total measure of the cover is <= 2 (<= because some intervals may overlap), so your "S" is of infinite measure, hence uncountable (although it does not contain any open interval).
Cheers, L.

