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.