Information on the thesis

The thesis is devoted to the Baire and Lebesgue classifications of maps of

one and several variables and to problems of extension of maps from different

functional classes. In particular, a new notions of σ-strongly functionally

discrete map and adhesive space are introduced in the thesis and applied

to a generalization of the classical Lebesgue-Hausdorff-Banach result. A

characterization of right and left B_α -compositors is given. A new concept

of almost strongly zero-dimensional space is introduced and a description

of such spaces in terms of Baire’s and Lebesgue’s classes is obtained.

Applying new notions of a convex combination, λ-sum of elements of an

equiconnected space and a PP-space we get general theorems on the Baire

and Lebesgue classification of vertically almost separately continuous functions.

The Kuratowski Extension Theorem is generalized in terms of α-embedded sets

introduced by the author. A notion of H₁ -retract is given and a characterization

of H₁ -retracts of completely metrizable spaces is obtained. We generalize the

Kuratowski-Sierpi´nski theorem on the first class functions with a connected

graph using a new notion of an A-closed set. New results on a dependence of

continuous functions defined on invariant subsets of uncountable products from

countably many coordinates is proved. Pointwise limits of finitely determined

continuous functions are investigated.

Keywords: Baire classification; Lebesgue classification; B_α -compositors; σ-

strongly functionally discrete map; adhesive space; strong PP-space; strongly

separately continuous map; H₁ -retract; B₁ -retract; weakly Gibson map; finitely

determined function.