Are scales Fr\'echet?
General Topology
2026-03-18 v2
Abstract
We continue the study of Dow spaces of a -scale, originally introduced by Alan Dow in "-Weight and the Fr\'echet-Urysohn property" (Topology and its Applications, Vol. 174, pp. 56-61). We prove that it is consistent that all such spaces are Fr\'echet, but it is also consistent that none of them is. We use these spaces to exhibit (consistently) a ideal that does not satisfy the Category Dichotomy. Finally, we prove that the Category Dichotomy holds for all co-analytic ideals.
Cite
@article{arxiv.2603.15567,
title = {Are scales Fr\'echet?},
author = {Raul Figueroa-Sierra and Osvaldo Guzmán and Michael Hrušák and Adam Kwela},
journal= {arXiv preprint arXiv:2603.15567},
year = {2026}
}