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A game for Baire's grand theorem

Lorenzo Notaro

TL;DR

The paper addresses characterizing $Baire class 1$ functions between separable metrizable spaces and connects this to a two-player topological game $G(f)$. It proves that Player II has a winning strategy in $G(f)$ iff $f$ is $Baire class 1$, and that Player I has a winning strategy iff there exists a compact $K subseteq X$ with $f|_K$ having no points of continuity. The determinacy of $G(f)$ for all $f$ is shown to be equivalent to a generalized Baire grand theorem, $GBT$, with $AC$ incompatible with $GBT$ but $AD$ and Solovay's model yielding both. The results connect topological function classes, determinacy, and Descriptive Set Theory principles, and discuss consistency strength and potential extensions beyond Polish domains.

Abstract

Generalizing a result of Kiss, we provide a game that characterizes Baire class 1 functions between arbitrary separable metrizable spaces. We show that the determinacy of our game is equivalent to a generalization of Baire's grand theorem, and that both these statements hold under AD and in Solovay's model.

A game for Baire's grand theorem

TL;DR

The paper addresses characterizing functions between separable metrizable spaces and connects this to a two-player topological game . It proves that Player II has a winning strategy in iff is , and that Player I has a winning strategy iff there exists a compact with having no points of continuity. The determinacy of for all is shown to be equivalent to a generalized Baire grand theorem, , with incompatible with but and Solovay's model yielding both. The results connect topological function classes, determinacy, and Descriptive Set Theory principles, and discuss consistency strength and potential extensions beyond Polish domains.

Abstract

Generalizing a result of Kiss, we provide a game that characterizes Baire class 1 functions between arbitrary separable metrizable spaces. We show that the determinacy of our game is equivalent to a generalization of Baire's grand theorem, and that both these statements hold under AD and in Solovay's model.
Paper Structure (3 sections, 7 theorems, 15 equations)

This paper contains 3 sections, 7 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. The game
  3. On the determinacy of $G(f)$

Key Result

Theorem 1

Let $X$ be a Polish space, $Y$ a separable metrizable space, and $f: X \rightarrow Y$. Then the following are equivalent:

Theorems & Definitions (23)

  • Theorem : Baire
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Claim 2.2.1
  • proof
  • Claim 2.2.2
  • proof
  • Claim 2.2.3
  • proof
  • ...and 13 more