Open filters and measurable cardinals

In this paper, we investigate the poset $\mathrm{OF}\left(X\right)$ of free open filters on a given space X. In particular, we characterize spaces for which $\mathrm{OF}\left(X\right)$ is a lattice. For each $n\in N$ we construct a scattered space X such that $\mathrm{OF}\left(X\right)$ 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 $\mathrm{OF}\left(X\right)$ 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 $\beta \left(\kappa \right)$ . Assuming the existence of n measurable cardinals, for every $m_0,\dots ,m_n\in N$ we construct a space X such that $\mathrm{OF}\left(X\right)$ is order isomorphic to ${\prod }_{i=0}^n{m}_i$ . 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.