Abstract In this paper, we investigate the poset OF (X) OF (X) of free open filters on a given space X. In particular, we characterize spaces for which OF (X) OF (X) is a lattice. For each n N n ∈ N we construct a scattered space X such that OF (X) OF (X) is order isomorphic to the n -element chain, which implies the affirmative answer to two questions of Mooney. Assuming CH we construct a scattered space X such that OF (X) OF (X) is order isomorphic to (+1, ) (ω + 1, ≥). To prove the latter facts we introduce and investigate a new stratification of ultrafilters which depends on scattered subspaces of () β (κ). Assuming the existence of n measurable cardinals, for every m₀, , m₍ N m 0, …, m n ∈ N we construct a space X such that OF (X) OF (X) is order isomorphic to ₈=₀ⁿmᵢ ∏ i = 0 n m i. Also, we show that the existence of a metric space possessing a free ₁ ω 1 -complete closed, G_ G δ, F F σ or Borel ultrafilter is equivalent to the existence of a measurable cardinal.
Bardyla et al. (Wed,) studied this question.