Are scales Fréchet?

Raul Figueroa-Sierra; Osvaldo Guzmán; Michael Hrušák; Adam Kwela

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arXiv · 2603.15567

Are scales Fréchet?

Raul Figueroa-Sierra·Osvaldo Guzmán·Michael Hrušák·Adam Kwela

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Abstract

We continue the study of Dow spaces of a $\mathfrak{b}$-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 $\triangle_{2}^{1}$ 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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Raul Figueroa-Sierra, Osvaldo Guzmán, Michael Hrušák, Adam Kwela. 2026-03-17. Are scales Fréchet?. https://arxiv.org/abs/2603.15567

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