# Open filters and measurable cardinals

## TL;DR

This paper explores mathematical structures called open filters and their properties in topological spaces, using concepts like measurable cardinals.

## Contribution

The paper introduces a new stratification of ultrafilters and provides constructions answering open questions in topology.

## Key findings

- A scattered space X can be constructed for each n ∈ N such that OF(X) is order isomorphic to an n-element chain.
- Assuming CH, a scattered space X exists where OF(X) is order isomorphic to (ω + 1, ≥).
- The existence of a metric space with certain ultrafilters is equivalent to the existence of a measurable cardinal.

## Abstract

In this paper, we investigate the poset \documentclass[12pt]{minimal}
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\begin{document}$$\textbf{OF}(X)$$\end{document}OF(X) of free open filters on a given space X. In particular, we characterize spaces for which \documentclass[12pt]{minimal}
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\begin{document}$$\textbf{OF}(X)$$\end{document}OF(X) is a lattice. For each \documentclass[12pt]{minimal}
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\begin{document}$$n\in \mathbb {N}$$\end{document}n∈N we construct a scattered space X such that \documentclass[12pt]{minimal}
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\begin{document}$$\textbf{OF}(X)$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$\textbf{OF}(X)$$\end{document}OF(X) is order isomorphic to \documentclass[12pt]{minimal}
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\begin{document}$$(\omega +1,\ge )$$\end{document}(ω+1,≥). To prove the latter facts we introduce and investigate a new stratification of ultrafilters which depends on scattered subspaces of \documentclass[12pt]{minimal}
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\begin{document}$$\beta (\kappa )$$\end{document}β(κ). Assuming the existence of n measurable cardinals, for every \documentclass[12pt]{minimal}
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\begin{document}$$m_0,\ldots ,m_{n}\in \mathbb {N}$$\end{document}m0,…,mn∈N we construct a space X such that \documentclass[12pt]{minimal}
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\begin{document}$$\textbf{OF}(X)$$\end{document}OF(X) is order isomorphic to \documentclass[12pt]{minimal}
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\begin{document}$$\prod _{i=0}^nm_i$$\end{document}∏i=0nmi. Also, we show that the existence of a metric space possessing a free \documentclass[12pt]{minimal}
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\begin{document}$$\omega _1$$\end{document}ω1-complete closed, \documentclass[12pt]{minimal}
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\begin{document}$$G_\delta $$\end{document}Gδ, \documentclass[12pt]{minimal}
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\begin{document}$$F_{\sigma }$$\end{document}Fσ or Borel ultrafilter is equivalent to the existence of a measurable cardinal.