The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters



Let X be an infinite set and let math formula and math formula denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by math formula the Stone space of the Boolean algebra of all subsets of X. We show:

  1. For every well-ordered cardinal number ℵ, math formula(ℵ) iff math formula(2).
  2. math formula iff “math formula is a continuous image of math formula” iff “math formula has a free open ultrafilter ” iff “every countably infinite subset of math formula has a limit point”.
  3. math formula implies “every open filter on math formula extends to an open ultrafilter” implies “math formulahas an open ultrafilter” implies math formula
  4. It is relatively consistent with math formula that math formula(ω) holds, whereas math formula(ω) fails. In particular, none of the statements given in (2) implies math formula(ω).