Boris A. Pasynkov

On an approach to constructing compacta with different dimensions dim and ind

Abstract In this paper we reveal common features in the constructions of some compacta X with dim X ind X . To any regular T 1 -space X a class T(X) of regular T 1 -spaces is related. Elements of T(X) are called tailings of X . Two concrete types of tailings are described. For any compactum X with dim X= ind X>0 , these constructions allow to obtain compacta Y with dim Y= dim X and ind Y> dim X . At the end of the paper, using tailings, we present compacta with different dimensions dim and ind constructed by means of product formation starting from compacta with equal dim and ind .

On metrizable type (MT-) maps and spaces

In this paper we define and study MT-maps, which are the fibrewise topological analogue of metrizable spaces, i.e., the extension of metrizability from the category Top to the category Top Y . Several characterizations and properties of MT-maps are proved. The notion of an MT-space as an MT-map preimage of a metrizable space is introduced. Examples of MT-spaces and their relation withM-spaces are given. Finally it is deduced that an MT-space with a G -diagonal is metrizable.

On the relative cellularity of Lindelöf subspaces of topological groups

Abstract The notions of the relative cellularity (≡ relative Souslin number) c ( X , Y ) and the relative τ -cellularity cel τ ( X , Y ) of a subspace X of a topological space Y are introduced. Always c ( X )⩾ c ( X , Y )⩽cel τ ( X , Y )⩽ cel τ ( X ). It is proved that cel τ ( X , G ) ⩽exp τ for any τ-Lindelof subspace X of any Hausdorff topological group G and c ( X )⩽cel τ ( X )⩽ exp τ if, in addition, X is a retract of G or even a retract of some G τ -subset of G . These results are deduced form the results concerning spaces with lattices of open mappings on them.

On the finite-dimensionality of topological products☆

Abstract It is proved that there exist integers e ( k , l ) ⩾ − 1 for k , l = − 1, 0, 1, … such that Ind X × Y ⩽ e (Ind X , Ind Y ) if the space X × Y is normal (and Hausdorff), Y is locally compact paracompact (in particular, compact) and Ind X Y k -space. Also it is proved that a strongly paracompact or a z -embedded subspace of a finitedimensional in the sense of Ind normal space is finite-dimensional in the same sense.

Tychonoff compactifications and R-completions of mappings and rings of continuous functions

All perfect extensions (with Tychonoff domains) of a continuous mapping f : X — Y between two Tychonoff spaces X and Y (in the category Top Y ) are described by means of presheaves of subrings of the rings C*(f -l U) where U is open in Y. In fact, a general description of all Tychonoff compactifications of a Tychonoff mapping f : X — Y is obtained. Our methods yield even a characterization of all Tychonoff compactifications of Tychonoff continuous images of f in the category Top Y .

Magill-type theorems for mappings

Magills and Rayburns theorems on the homeomorphism of Stone-Cech remainders and some of their generalizations to the remainders of arbitrary Hausdorff compactifications of Tychonoff spaces are extended to some class of mappings.

On inverse systems and cardinal functions of topological spaces

Abstract After proving a general theorem for some special inverse system, we introduce the class of openly generated spaces, which generalizes κ-metrizable compacta in the sense of Scepin as well as od-spaces in the sense of Uspenskii, which in their turn generalize topological groups and Dugunji compacta. It is proved in rather general supposition that an openly generated space X has countable ω-cellularity and is hereditarily perfectly κ-normal; the closure of any union of Gδ-sets is a Gδ-set in X; if besides X is completely paracompact then ind X = Ind X.

Some problems in the dimension theory of compacta

Publisher Summary This chapter discusses some problems in the dimension theory of compacta. All topological spaces considered in this chapter are assumed to be Tychonoff and called simply spaces; maps mean continuous maps of topological spaces. Almost all problems posed in the chapter concern compact spaces. The chapter defines the dimension Δ of a paracompact space X as: Δ X ≤ n if there exists a strongly zero-dimensional paracompact space X 0 and a surjective closed map f : X 0 → X such that | f − 1 x | ≤ n + 1 for any x ∈ X . It is known that the three basic dimension functions dim, ind, and Ind coincide for compact metrizable spaces, that is, dim X = ind X = Ind X for any compact metrizable space X . In 1936, Alexandroff asked whether they coincide for arbitrary compact spaces. In 1941, he proved that dim X ≤ ind X for any compact space X . Also Ind X ≤ Ind X for any normal space X and Ind X ≤ Δ X for any paracompact space X. The chapter discusses about on the coincidence of dim, ind, Ind, and Δ for compact spaces. It explains noncoincidence of dim and ind for compact spaces as well as noncoincidence of ind and Ind for compact spaces. A detailed discussion on dimensional properties of topological products is also presented.