Open filters and measurable cardinals
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911931135164416 |
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| author | Bardyla, Serhii Supina, Jaroslav Zdomskyy, Lyubomyr |
| author_facet | Bardyla, Serhii Supina, Jaroslav Zdomskyy, Lyubomyr |
| contents | 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 $(ω+1,\geq)$. To prove the latter facts we introduce and investigate a new stratification of ultrafilters which depends on scattered subspaces of $β(κ)$. 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 $ω_1$-complete closed, $G_δ$, $F_σ$ or Borel ultrafilter is equivalent to the existence of a measurable cardinal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_08704 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Open filters and measurable cardinals Bardyla, Serhii Supina, Jaroslav Zdomskyy, Lyubomyr General Topology 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 $(ω+1,\geq)$. To prove the latter facts we introduce and investigate a new stratification of ultrafilters which depends on scattered subspaces of $β(κ)$. 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 $ω_1$-complete closed, $G_δ$, $F_σ$ or Borel ultrafilter is equivalent to the existence of a measurable cardinal. |
| title | Open filters and measurable cardinals |
| topic | General Topology |
| url | https://arxiv.org/abs/2301.08704 |