Categorical concepts and their generalization by means of game theory (german: Kategorien-Konzepte und ihre spieltheoretische Verallgemeinerung)

TL;DR

This paper explores the generalization of size concepts in topology and set theory using game theory, particularly Banach-Mazur games, under the Axiom of Determinacy, providing new proofs and definability results.

Contribution

It introduces a game-theoretic framework for size concepts like Baire category and sigma-category, extending classical theorems with new proofs under AD.

Findings

Generalized Banach-Mazur games characterize size concepts.

Proved analogues of Cantor-Bendixson and Heine-Borel theorems.

Established definability results for size concepts.

Abstract

This paper deals with different concepts for characterizing the size of mathematical objects. A game theoretic investigation and generalization of two size concepts, which can both be formulated in topological terms, is provided: the so called "Baire category" and the "$\sigma $-category". This is mainly done by means of (generalized) Banach-Mazur games using the Axiom of determinacy (the inconsistency of AC and AD is reflected in the beginning and a weaker form of AC is chosen for proofs). Analogue versions of Cantor-Bendixson and Heine-Borel are proofed as well as some definability results. Such size concepts are established f.i. in measure and integration theory, set theory and topology often leading to mathematically precise formulations of "fuzziness". In measure and integration theory f.i. one defines for a "measure space" $(\Omega, \underline{A}, \mu)$ - i.e. $\Omega$ is a non empty set, $\underline{A}$ a $\sigma$-algebra on $\Omega$ and $\mu$ a measure on $(\Omega,\underline{A})$ - and a property $E \subset \Omega$ of elements of $\Omega$, that the property $E$ is valid in a set $A\in\underline{A}$ "$\mu$-almost everywhere", if $E$ is true in $A$ up to a "$\mu$-null set" - i.e. $\exists N\in\underline{A}{ } (\mu(N)=0 \wedge A\cap N^\complement \subset E)$. This is crucial for the formulation of uniqueness statements in measure and integration theory as its theorems usually only apply up to "small" ($\mu$-null) sets. Key words: Axiom of choice AC, Axiom of determinacy AD, Baire category theorem, Baire property, Baire space, Banach-Mazur games, Borel hierachy, Cantor-Bendixson, definability, Gale-Stewart, Heine-Borel, Lusin hierarchy, meager sets, perfect set property, polish spaces, projective hierarchy, sigma bounded sets, sigma compact sets, superperfect sets, topological games, winning strategy