# Maslyuchenko O. V.

### Properties of the Ceder Product

Maslyuchenko O. V., Maslyuchenko V. K., Myronyk O. D.

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 780-787

We study properties of the Ceder product $X ×_b Y$ of topological spaces $X$ and $Y$, where $b ∈ Y$, recently introduced by the authors. Important examples of the Ceder product are the Ceder plane and the Alexandroff double circle. In particular, for $i = 0, 1, 2, 3$ we establish necessary and sufficient conditions for the Ceder product to be a $T_i$ -space. We prove that the Ceder product $X ×_b Y$ is metrizable if and only if the spaces $X$ and $\overset{.}{Y}=Y\backslash \left\{b\right\}$ are metrizable, $X$ is $σ$-discrete, and the set $\{b\}$ is closed in $Y$. If $X$ is not discrete, then the point $b$ has a countable base of closed neighborhoods in $Y$.

### Theorems on decomposition of operators in *L*_{1} and their generalization to vector lattices

Maslyuchenko O. V., Mykhailyuk V. V., Popov M. M.

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 26-35

The Rosenthal theorem on the decomposition for operators in *L*_{1} is generalized to vector lattices and to regular operators on vector lattices. The most general version turns out to be relatively simple, but this approach sheds new light on some known facts that are not directly related to the Rosenthal theorem. For example, we establish that the set of narrow operators in *L*_{1} is a projective component, which yields the known fact that a sum of narrow operators in *L*_{1} is a narrow operator. In addition to the Rosenthal theorem, we obtain other decompositions of the space of operators in *L*_{1}, in particular the Liu decomposition.

### Direct and Inverse Problems of Baire Classification of Integrals Depending on a Parameter

Banakh T. O., Kutsak S. M., Maslyuchenko O. V., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1443-1457

We study the problem of the Baire classification of integrals *g* (*y*) = (*If*)(*y*) = ∫ _{X} *f*(*x, y*)*d*μ(*x*), where *y* is a parameter that belongs to a topological space *Y* and *f* are separately continuous functions or functions similar to them. For a given function *g*, we consider the inverse problem of constructing a function *f* such that *g* = *If*. In particular, for compact spaces *X* and *Y* and a finite Borel measure μ on *X*, we prove the following result: In order that there exist a separately continuous function *f* : *X* × *Y* → ℝ such that *g* = *If*, it is necessary and sufficient that all restrictions *g*|_{ Y } _{ n } of the function *g*: *Y* → ℝ be continuous for some closed covering { *Y* _{ n } *: n* ∈ ℕ} of the space *Y*.

### Separately continuous functions with respect to a variable frame

Herasymchuk V. H., Maslyuchenko O. V., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1281-1286

We show that the set *D*(*f*) of discontinuity points of a function *f* : **R** ^{2} → **R** continuous at every point *p* with respect to two variable linearly independent directions *e* _{1}(*p*) and *e* _{2}(*p*) is a set of the first category. Furthermore, if *f* is differentiable along one of directions, then *D*(*f*) is a nowhere dense set.

### Construction of a separately continuous function with given oscillation

Maslyuchenko O. V., Maslyuchenko V. K.

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 948–959

We investigate the problem of construction of a separately continuous function *f* whose oscillation is equal to a given nonnegative function *g*. We show that, in the case of a metrizable Baire product, the problem under consideration is solvable if and only if *g* is upper semicontinuous and its support can be covered by countably many sets, which are locally contained in products of sets of the first category.