On some new ideals on the Cantor and Baire spaces

Abstract

Abstract. We define and investigate some new ideals of subsets of the Cantor space and the Baire space. We show that combinatorial properties of these ideals can be described by the splitting and reaping cardinal numbers. We show that there exist perfect Luzin sets for these ideals on the Baire space. IntroductionFor each infinite subset T of the set ω of all natural numbers let us denote by K(T ) the σ-ideal of meagre subsets of the space 2 T with the canonical product topology. By L(T ) we denote the σ-ideal of Lebesgue measure zero subsets of 2 T with respect to the canonical product measure.Notice that if T is a subset of ω then we can indentify the spaces 2 T × 2 ω\T and the Cantor space 2 ω using the canonical homeomorphism π T defined by π T (x) = (x|T, x|(ω \ T )). Directly from the definition of meagre sets it follows that if A ∈ K(T ), then A × 2 ω\T ∈ K(ω). The same observation is also true for the ideal L(ω), but it is evidently false for the σ-ideal of all countable subsets of the Cantor space. We call this property of the ideals K(ω) and L(ω) productivity.There are other natural productive σ-ideals of subsets of the Cantor space, e.g. the σ-ideal K(ω) ∩ L(ω). It is interesting that among them there exists the least productive σ-ideal which contains all points. We call this ideal S 2 . There exists also the least productive ideal of subsets of 2 ω , and we call it I 2 . These ideals have Borel bases but they do not satisfy the countable chain condition-there exists a family of continuum many pairwise disjoint Borel sets outside the ideal S 2 . The ideal I 2 is not σ-additive, and the ideal S 2 is precisely σ-additive. The minimum cardinality of bases of these ideals is continuum. The covering number cov of both ideals is equal to the reaping cardinal r, and the last basic combinatorial invariants (cardinal numbers non) are described in terms of splitting cardinal numbers. These results show that both splitting and reaping cardinals are closely connected with natural mathematical objects on the classical Cantor space.We also consider the minimal productive σ-ideal S ω of subsets of the Baire space

On critical cardinalities related to $Q$-sets

“…For the compact Polish group Z ω 2 = {0, 1} ω the ideal I ccc (Z ω 2 ), denoted by I ccc , was introduced and studied by Zakrzewski [46], [47] who proved that s ω ≤ non(I ccc ) ≤ min{non(M), non(N )}. Here s ω is the ω-splitting number introduced in [30] and studied in [11], [27]. We recall that…”

Transitive Properties of the Ideal 𝕊₂

“…This ideal appeared for the first time in [10], but only incidentally. It was thoroughly investigated by Cichoń and Kraszewski in [5]. It turned out that cardinal characteristics of S 2 are strongly connected with some intensively studied combinatorial properties of subsets of natural numbers (the splitting and reaping numbers).…”

“…Let S 2 denote the σ-ideal of subsets of 2 ω , which is generated by the family {[f ] : f ∈ P if }. We recall some properties of S 2 , which were proved in [5]. We call a family F ⊆ P if normal if for each two different f 1 , f 2 ∈ F we have dom(f 1 ) ∩ dom(f 2 ) = ∅.…”

“…More about cardinal numbers connected with the relation of splitting can be found in [8]. It was proved in [5] that non(S 2 ) = ℵ 0 -s.…”

Properties of ideals on the generalized Cantor spaces

“…However, every a -ideal has its "productive closure". [3]. Earlier it appeared in [13], but only incidentally.…”