2023/01/20 by Serhii Bardyla, Bardyla, Serhii, Jaroslav Šupina +3 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2301.08704
openalex publication_date 2023/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we investigate the poset OF(X) of free open filters on a given space X. In particular, we characterize spaces for which OF(X) is a lattice. For each n∈ℕ we construct a scattered space X such that 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 OF(X) is order isomorphic to (ω+1,≥). 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 m0,…,mn∈\mathbb N we construct a space X such that OF(X) is order isomorphic to ∏i=0nmi. Also, we show that the existence of a metric space possessing a free ω1-complete closed, Gδ, Fσ or Borel ultrafilter is equivalent to the existence of a measurable cardinal.