Bernstein sets andκ-coverings

Abstract

Abstract. In this paper we study a notion of a κ-covering in connection with Bernstein sets and other types of nonmeasurability. Our results correspond to those obtained by Muthuvel in [7] and Nowik in [8]. We consider also other types of coverings. Definitions and notationIn 1993 Carlson in his paper [3] introduced a notion of κ-coverings and used it for investigating whether some ideals are or are not κ-translatable. Later on κ-coverings were studied by other authors, e.g. Muthuvel (cf. [7]) and Nowik (cf.[8], [9]). In this paper we present new results on κ-coverings in connection with Bernstein sets. We also introduce two natural generalizations of the notion of κ-coverings, namely κ-S-coverings and κ-I-coverings.We use standard set-theoretical notation and terminology from [1]. Recall that the cardinality of the set of all real numbers R is denoted by c. The cardinality of a set A is denoted by |A|. If κ is a cardinal number then

Bernstein sets with algebraic properties

“…In the following section, we will construct Bernstein sets being κ-coverings and consider the possibility of constructing partitions of R into such sets. Our results improve and complement results of [4], as well as answer an open question posed there. Later, we improve results from [8], where nonmeasurable (in the sense of Lebesgue measurability and Baire property) sets with certain algebraic properties were constructed -we construct Bernstein sets with the same properties.…”

“…The first one will be useful later in one of our constructions. The analogous method to our approach was used in [4], but the fact was not formulated in such a general setting. We construct a partition {C α : α < κ} by transfinite induction.…”

“…The previous theorem provides us just one Bernstein covering, while the authors of [4] were interested in constructing partitions into such coverings. If c is regular, Theorem 2.1 is in a sense the best possible -we have as many sets in the partition as possible and their covering condition is as strong as possible.…”

Two point sets with additional properties

“…We recall the notion of completely I-nonmeasurability which was studied e.g. in [3], [7], [14], [15], [18]. This notion is also known as the I-Bernstein set.…”

“…(A is a κ-covering if for every subset X of size κ there exists a translation h of R 2 such that h[X] ⊆ A; A is a κ-I-covering if for every subset X of size κ there exists an isomorphism h of R 2 such that h[X] ⊆ A; see [7]. )…”