Year: 2025
Type: article
Abstract: 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 ∈ 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 , ≥ ) . 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 , … , m n ∈ N we construct a space X such that $$\textbf{OF}(X)$$ OF ( X ) is order isomorphic to $$\prod _{i=0}^nm_i$$ ∏ 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
Source: Archive for Mathematical Logic
Institutions: University of Vienna, University of Pavol Jozef Šafárik, TU Wien
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4414272484