Forcing, games and families of closed sets

Marcin Sabok

paper

Forcing, games and families of closed sets

arXiv:0910.2318

Abstract

We propose a new, game-theoretic, approach to the idealized forcing, in terms of fusion games. This generalizes the classical approach to the Sacks and the Miller forcing. For definable ( $Π_{1}^{1}$ on $Σ_{1}^{1})$ σ $- i d e a l s w es h o w t ha t i f a$ σ $- i d e a l i s g e n er a t e d b y c l ose d se t s, t h e ni t i s g e n er a t e d b y c l ose d se t s ina l l f or c in g e x t e n s i o n s . W e a l so p r o v e anin f ini t e - d im e n s i o na l v er s i o n o f t h e S o l ec k i d i c h o t o m y f or ana l y t i cse t s . A m o n g e x am pl es, w e in v es t i g a t e t h e$ σ $- i d e a l$ \E $g e n er a t e d b y c l ose d n u l l se t s an d$ σ$-ideals connected with not piecewise continuous functions.