On Borel sets belonging to every invariant ccc $\sigma $-ideal on $2^{\mathbb {N}}$

On Borel Sets Belonging to Every Invariant Ccc Σ-ideal on 2n

Let Iccc be the σ-ideal of subsets of the Cantor group 2N generated by Borel sets which belong to every translation invariant σ-ideal on 2N satisfying the countable chain condition (ccc). We prove that Iccc strongly violates ccc. This generalizes a theorem of Balcerzak-Rosłanowski-Shelah stating the same for the σ-ideal on 2N generated by Borel sets B ⊆ 2N which have perfectly many pairwise dis...

On invariant sets topology

In this paper, we introduce and study a new topology related to a self mapping on a nonempty set.Let X be a nonempty set and let f be a self mapping on X. Then the set of all invariant subsets ofX related to f, i.e. f := fA X : f(A) Ag P(X) is a topology on X. Among other things,we nd the smallest open sets contains a point x 2 X. Moreover, we find the relations between fand To f . For insta...

A Course on Borel Sets

Theorems on Sets Not Belonging to Algebras

Let A1, . . . ,An,An+1 be a finite sequence of algebras of sets given on a set X, ⋃n k=1Ak 6= P(X), with more than 4 3 n pairwise disjoint sets not belonging to An+1. It has been shown in the author’s previous articles that in this case ⋃n+1 k=1 Ak 6= P(X). Let us consider, instead of An+1, a finite sequence of algebras An+1, . . . ,An+l. It turns out that if for each natural i ≤ l there exist ...

On Infinitary Rational Relations and Borel Sets

We prove in this paper that there exists some infinitary rational relations which are Σ03-complete Borel sets and some others which are Π03-complete. These results give additional answers to questions of Simonnet [Sim92] and of Lescow and Thomas [Tho90] [LT94].