Are scales Fr\'echet?

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 $\mathfrak{b}$-scale, originally introduced by Alan Dow in "$\pi$-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 $\triangle_{2}^{1}$ ideal that does not satisfy the Category Dichotomy. Finally, we prove that the Category Dichotomy holds for all co-analytic ideals.