Open filters and measurable cardinals

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In this paper, we investigate the poset textbf{OF}(X) of free open filters on a given space X. In particular, we characterize spaces for which textbf{OF}(X) is a lattice. For each nin mathbb {N} we construct a scattered space X such that textbf{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 textbf{OF}(X) is order isomorphic to (omega +1,ge ). To prove the latter facts we introduce and investigate a new stratification of ultrafilters which depends on scattered subspaces of beta (kappa ). Assuming the existence of n measurable cardinals, for every m_0,ldots ,m_{n}in mathbb {N} we construct a space X such that textbf{OF}(X) is order isomorphic to prod _{i=0}^nm_i. Also, we show that the existence of a metric space possessing a free omega _1-complete closed, G_delta , F_{sigma } or Borel ultrafilter is equivalent to the existence of a measurable cardinal.