# '22

36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

### "Open filters and measurable cardinals"

#### Bardyla, Serhii

In this presentation we discuss the poset $OF(X)$ of all free open filters on a given Hausdorff space $X$. We characterize spaces whose posets of free open filters are lattices. For each $n\in\mathbb N$ we construct a scattered space $X$ such that $OF(X)$ is order isomorphic to the $n$-element chain. This result answers two questions of Mooney. It is proved that the existence of a space $X$ which possesses a free $\omega_1$-complete open ultrafilter is equivalent to the existence of a measurable cardinal. This provides an answer to an old question of Liu. Assuming the existence of $n$ measurable cardinals, we construct a space $X$ such that $OF(X)$ is isomorphic to $2^{n+1}$. Our research motivates a new stratification of ultrafilters on $\kappa$ which is defined with a help of scattered subspaces of $\beta(\kappa)$. Some properties (related to measurable cardinals) of this stratification will be revealed. This is a joint work with J. Šupina and L. Zdomskyy.

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