Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show:
- For every well-ordered cardinal number ℵ, (ℵ) iff (2ℵ).
- iff “ is a continuous image of ” iff “ has a free open ultrafilter ” iff “every countably infinite subset of has a limit point”.
- implies “every open filter on extends to an open ultrafilter” implies “has an open ultrafilter” implies
- It is relatively consistent with that (ω) holds, whereas (ω) fails. In particular, none of the statements given in (2) implies (ω).