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Are scales Fréchet?

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

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.

Are scales Fréchet?

Abstract

We continue the study of Dow spaces of a -scale, originally introduced by Alan Dow in "-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 ideal that does not satisfy the Category Dichotomy. Finally, we prove that the Category Dichotomy holds for all co-analytic ideals.
Paper Structure (13 sections, 32 theorems, 10 equations)

This paper contains 13 sections, 32 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries on cardinal invariants of the continuum
  4. Preliminaries on filters and ideals
  5. Preliminaries on Topology
  6. Preliminaries on Forcing
  7. Preliminaries on Forcing and topology
  8. The Dow space of a $\mathfrak{b}$-scale
  9. All $\mathfrak{b}$-scales may be Fréchet
  10. There may be a non-Fréchet $\mathfrak{b}$-scale
  11. There may be no Fréchet $\mathfrak{b}$-scale
  12. The Category Dichotomy for definable ideals
  13. Open Questions

Key Result

Theorem 2

It is consistent that every countable, Fréchet topological group is second countable.

Theorems & Definitions (66)

  • Theorem 2: H., Ramos MalykhinProblem
  • Theorem 3: Dow PiWeightandFrechetProperty
  • Theorem 4
  • Lemma 5
  • Proposition 6: Baumgartner, Dordal BaumgartnerDordalAdjoining
  • proof
  • Lemma 7
  • proof
  • Definition 8
  • Definition 9
  • ...and 56 more