In this paper, we investigate the poset <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textbf{OF}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">OF</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of free open filters on a given space X. In particular, we characterize spaces for which <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textbf{OF}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">OF</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a lattice. For each <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> we construct a scattered space X such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textbf{OF}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">OF</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textbf{OF}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">OF</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is order isomorphic to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(\omega +1,\ge )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>≥</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. To prove the latter facts we introduce and investigate a new stratification of ultrafilters which depends on scattered subspaces of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\beta (\kappa )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo stretchy="false">(</mo> <mi>κ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Assuming the existence of n measurable cardinals, for every <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$m_0,\ldots ,m_{n}\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mn>0</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>m</mi> <mi>n</mi> </msub> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> we construct a space X such that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textbf{OF}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">OF</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is order isomorphic to <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\prod _{i=0}^nm_i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∏</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>n</mi> </msubsup> <msub> <mi>m</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. Also, we show that the existence of a metric space possessing a free <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\omega _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-complete closed, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$G_\delta $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mi>δ</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$F_{\sigma }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>σ</mi> </msub> </math></EquationSource> </InlineEquation> or Borel ultrafilter is equivalent to the existence of a measurable cardinal.

Open filters and measurable cardinals

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2025年9月17日

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Serhii Bardyla,
Jaroslav Šupina,
Lyubomyr Zdomskyy

摘要

In this paper, we investigate the poset $\textbf{OF}(X)$ $OF (X)$ of free open filters on a given space X. In particular, we characterize spaces for which $\textbf{OF}(X)$ $OF (X)$ is a lattice. For each $n\in \mathbb {N}$ $n \in N$ we construct a scattered space X such that $\textbf{OF}(X)$ $OF (X)$ 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 $\textbf{OF}(X)$ $OF (X)$ is order isomorphic to $(\omega +1,\ge )$ $(ω + 1, \geq)$ . To prove the latter facts we introduce and investigate a new stratification of ultrafilters which depends on scattered subspaces of $\beta (\kappa )$ $β (κ)$ . Assuming the existence of n measurable cardinals, for every $m_0,\ldots ,m_{n}\in \mathbb {N}$ $m_{0}, \dots, m_{n} \in N$ we construct a space X such that $\textbf{OF}(X)$ $OF (X)$ is order isomorphic to $\prod _{i=0}^nm_i$ $\prod_{i = 0}^{n} m_{i}$ . Also, we show that the existence of a metric space possessing a free $\omega _1$ $ω_{1}$ -complete closed, $G_\delta $ $G_{δ}$ , $F_{\sigma }$ $F_{σ}$ or Borel ultrafilter is equivalent to the existence of a measurable cardinal.