Open filters and measurable cardinals

TL;DR

This paper explores the structure of free open filters on spaces, characterizing when they form lattices, constructing spaces with specific filter posets, and linking ultrafilter stratifications to measurable cardinals.

Contribution

It introduces a new stratification of ultrafilters, characterizes spaces with lattice filter posets, and connects ultrafilter properties to the existence of measurable cardinals.

Findings

Constructed spaces with filter posets as finite chains and $( ext{omega}+1, ext{geq})$

Introduced a new ultrafilter stratification based on scattered subspaces

Linked ultrafilter existence in metric spaces to 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 $(\omega+1,\geq)$. 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 $\mathbf{OF}(X)$ 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.