In article <Pine.NEB.email@example.com>, William Elliot <firstname.lastname@example.org> 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).