Are scales Fréchet?

2026/03/16 by Raul Figueroa-Sierra, Raúl Figueroa-Sierra, Osvaldo Guzmán +2 · 1 voice
Mathematics · #math.GN

paper · pdf · doi:10.1016/j.topol.2026.109901

published in Topology and its Applications 391, 109901 (Elsevier BV)

arxiv published 2026/03/16 · arxiv updated 2026/03/17 · crossref created 2026/07/07 · crossref deposited 2026/08/06 · crossref indexed 2026/08/06 · crossref issued 2026/11/01 · crossref published 2026/11/01 · crossref published-print 2026/11/01

Abstract

We continue the study of Dow spaces of a \mathfrakb-scale, originally introduced by Alan Dow in "π-Weight and the Fréchet-Urysohn property" (Topology and its Applications, Vol. 174, pp. 56-61). We prove that it is consistent that all such spaces are Fréchet, but it is also consistent that none of them is. We use these spaces to exhibit (consistently) a \triangle₂¹ ideal that does not satisfy the Category Dichotomy. Finally, we prove that the Category Dichotomy holds for all co-analytic ideals.

Citations

• Some results on the π-weight of countable Fréchet-Urysohn spaces
• New examples in the study of selectively separable spaces
• The Category Dichotomy for Ideals
• Invariant Ideal Axiom
• Projective well-orders and coanalytic witnesses
• Coanalytic Ultrafilter Bases
• Katětov order on Borel ideals
• Topological representations
• Transfinite inductions producing coanalytic sets
• Cofinality spectrum theorems in model theory, set theory and general topology
• Unbounded and dominating reals in Hechler extensions
• Borel Conjecture and Dual Borel Conjecture
• Combinatorial properties of Hechler forcing
• Classical Descriptive Set Theory

Discussions

• Raul Figueroa-Sierra, Osvaldo Guzm\'an, Michael Hru\v{s}\'ak, Adam Kwela: Are scales Fr\'echet? https://arxiv.org/abs/2603.15567 https://arxiv.org/pdf/2603.15567 https://arxiv.org/html/2603.15567 [bsky, 0 points, 0 comments]

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• The Category Dichotomy for Ideals
• Beyond Lebesgue and Baire IV: Density topologies and a converse Steinhaus-Weil Theorem
• Diametral dimensions of Frechet spaces
• The BSE property for vector-valued Frechet Lipschitz algebras
• Ideal approach to convergence in functional spaces