Are scales Fr\'echet?
Raul Figueroa-Sierra, Osvaldo Guzm\'an, Michael Hru\v{s}\'ak, Adam Kwela

TL;DR
This paper investigates the conditions under which Dow spaces of a $$-scale are Fréchet, showing consistency results and exploring their implications for ideals and the Category Dichotomy.
Contribution
It proves the consistency that all Dow spaces of a $$-scale are Fréchet and that none are, and establishes the Category Dichotomy for co-analytic ideals.
Findings
All such spaces can be Fréchet under certain set-theoretic assumptions.
Existence of a $ riangle_{2}^{1}$ ideal that does not satisfy the Category Dichotomy.
The Category Dichotomy holds for all co-analytic ideals.
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.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Rings, Modules, and Algebras
