Vitalij A. Chatyrko

Transfinite extension of Steinke's dimension

We introduce a transfinite extension trt of dimension t from [10]. We extend all main theorems for t such as the sum theorems, the product theorem and the compactification theorem to the transfinite numbers. We also consider the class of all spaces X with trt (X) = ω0, the so called finite-dimensional separated spaces [2], with respect to the usual classes of infinite-dimensional spaces.

Continuum many Frechet types of hereditarily strongly infinite-dimensional Cantor manifolds

In this note we construct a family of continuum many hereditarily strongly infinite-dimensional Cantor manifolds such that for every two spaces from this family, no open subset of one is embeddable into the other.

On the families of sets without the Baire property generated by the Vitali sets

Let A be the family of all meager sets of the real line ℝ, V be the family of all Vitali sets of ℝ, V1 be the family of all finite unions of elements of V and V2 = {(C \ A1) ∪ A2: C ∈ V1; A1, A2 ∈ A}. We show that the families V, V1, V2 are invariant under translations of ℝ, and V1, V2 are abelian semigroups with the respect to the operation of union of sets. Moreover, V ⊂ V1 ⊂ V2 and V2 consists of zero-dimensional sets without the Baire property. Then we extend the results above to the Euclidean spaces ℝn, n ≥ 2, and their products with the finite powers of the Sorgenfrey line.

On countable unions of nonmeager sets in hereditarily Lindelöf spaces

It is well known that any Vitali set on the real line ℝ does not possess the Baire property. The same is valid for finite unions of Vitali sets. What can be said about infinite unions of Vitali sets? Let S be a Vitali set, Sr be the image of S under the translation of ℝ by a rational number r and F = {Sr: r is rational}. We prove that for each non-empty proper subfamily F′ of F the union ∪F′ does not possess the Baire property. We say that a subset A of ℝ possesses Vitali property if there exist a non-empty open set O and a meager set M such that A ⊃ O \ M. Then we characterize those non-empty proper subfamilies F′ of F which unions ∪F′ possess the Vitali property.

Classical dimension theory

Publisher Summary This chapter presents an overview of the classical dimension theory. It recalls results obtained after 1990 and consider problems that could probably be solved without the use of deep methods of algebra or geometry. All spaces considered are assumed to be regular T 1 . The covering dimension of a completely regular space X , dim X , is defined as: Dim X ≤ n if each finite cover of X by functionally open sets has a finite refinement by functionally open sets such that each point belongs to at most n + 1 of them. The large inductive dimension of a space X, Ind X, is defined inductively by the following way. Ind X = − 1 if and only if X = ϕ . Ind X ≤ n if every closed subset A of X has arbitrarily tight open neighborhoods U with Ind Bd U ≤ n − 1 , where Bd C denotes the boundary of a set C . (If X is normal then, equivalently, the set A can be separated from the complement of U by a partition C with Ind C ≤ n − 1 .) It is also possible to get the definition of the small inductive dimension of a space X , ind X , from the definition of Ind X by replacing the set A by a point.

On conforming axiomatics

We construct conforming axiomatics of the covering dimension dim and the D-dimension (Henderson, 1968) on the class of all metrizable compacta.

Dimension axioms and decomposing mappings

An axiomatic definition of the covering dimension dim in the class of all (closed) subsets of finite-dimensional cubes is given relative to decomposing mappings. An axiomatic definition of the possible transfinite extension of this dimension in the class of all (closed) subsets of the Smirnov compacta is suggested. Bibliography: 15 titles.