Open filters and measurable cardinals

Abstract

In this paper, we investigate the poset $\mathbf{OF}(X)$ of free open filters on a given space $X$. In particular, we characterize spaces for which $\mathbf{OF}(X)$ is a lattice. For each $n\in \mathbb{N}$ we construct a scattered space $X$ such that $\mathbf{OF}(X)$ 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 $\mathbf{OF}(X)$ 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 $m_0,\dots ,m_n\in \mathbb{N}$ we construct a space $X$ such that $\mathbf{OF}(X)$ 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.