N - Localization Property

The present paper is concerned with the n–localization property and its preservation in countable support (CS) iterations. This property was first introduced in Newelski and Ros lanowski [10, p. 826]. Definition 0.1. Let n be an integer greater than 1. (1) A tree T is an n–ary tree provided that (∀s ∈ T)(|succ T (s)| ≤ n). (2) A forcing notion P has the n–localization property if P " ∀f ∈ ω ω ∃T ∈ V T is an n–ary tree and f ∈ [T ] ". In [10, Theorem 2.3] we showed that countable support products of the n–Sacks forcing notion D n (see Definition 1.5(1) here) have the n–localization property. That theorem was used to obtain some consistency results concerning cardinal characteristics of the ideal determined by unsymmetric games. Soon after this, the uniform n–Sacks forcing notion Q n (see Definition 1.5(2)) was introduced in [11, §4] and applied in the proof of [11, Theorem 5.13]. The crucial property of Q n which was used there is that the CS iterations of Q n have the n–localization property, but in [11] we only stated that the proof is similar to that of [10, Theorem 2.3]. One of the difficulties with the n–localization property was that there was no " preservation theorem " for it. Geschke and Quickert [5] give full and detailed proofs of the 2–localization property for both CS products and CS iterations of the Sacks forcing D 2 (and those proofs can be easily rewritten for n–localization property and D n). And the same proof can be repeated for Q n , but a more general theorem has been missing. Recently, the n–localization property, the σ–ideal generated by n–ary trees and n–Sacks forcing notion D n have been found applicable to some questions concerning convexity numbers of closed subsets of R n , see Geschke, Kojman, Kubi´s and Schipperus [4], Geschke and Kojman [3] and most recently Geschke [2]. The latter paper is raison d'ˆ etre for this note — when I read [2] to write a review for Mathematical Reviews I wanted to check as many technical details as I could. In [2, §2] an interesting forcing notion 1 P G was introduced and a proof was given that it has the n–localization property. However, the proof that the CS iteration of this forcing has the n–localization property was left to the reader as " …