Are scales Fréchet?

Abstract

We continue the study of Dow spaces of a $\mathfrak{b}$-scale, originally introduced by Alan Dow in "$\ddot{I}€$-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 ${△}_2^1$ ideal that does not satisfy the Category Dichotomy. Finally, we prove that the Category Dichotomy holds for all co-analytic ideals.