# Mykhailyuk V. V.

### Upper and lower Lebesgue classes of multivalued functions of two variables

Karlova O. O., Mykhailyuk V. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1097-1106

We introduce a functional Lebesgue classification of multivalued mappings and obtain results on upper and lower Lebesgue classifications of multivalued mappings $F : X \times Y \multimap Z$ for wide classes of spaces $X, Y$ and $Z$.

### Cross Topology and Lebesgue Triples

Karlova O. O., Mykhailyuk V. V.

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 722–727

The cross topology γ on the product of topological spaces *X* and *Y* is the collection of all sets *G* ⊆ *X* × *Y* such that the intersections of *G* with every vertical line and every horizontal line are open subsets of the vertical and horizontal lines, respectively. For the spaces *X* and *Y* from a class of spaces containing all spaces \( {{\mathbb{R}}^n} \) , it is shown that there exists a separately continuous function *f* : *X* × *Y* → (*X* × *Y*, γ) which is not a pointwise limit of a sequence of continuous functions. We also prove that each separately continuous function is a pointwise limit of a sequence of continuous functions if it is defined on the product of a strongly zero-dimensional metrizable space and a topological space and takes values in an arbitrary topological space.

### Joint continuity of $K_h C$-functions with values in moore spaces

Filipchuk O. I., Maslyuchenko V. K., Mykhailyuk V. V.

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1539 – 1547

We introduce a notion of a categorical cliquish mapping and prove that, for each $K_h C$-mapping $f : X \times Y \rightarrow Z$ (here, $X$ is a topological space, $Y$ is a first countable space, and $Z$ is a Moore space) with categorical cliquish horizontal $y$-sections $f_y$ , the sets $C_y (f)$ are residual $G_\delta$-sets in $X$ for each $y \in Y.$

### Weak local homeomorphisms and B-favorable spaces

Karlova O. O., Mykhailyuk V. V.

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1189–1195

Let *X* and *Y* be topological spaces such that every mapping *f* : *X* → *Y* for which the set *f *^{ - 1}(*G*)
is an *f *_{σ} -set in *X* for any set *G* open in *Y* can be represented as a pointwise limit of continuous mappings *f _{n}* :

*X*→

*Y*. The question of subspaces

*Z*of the space

*Y*for which mappings

*f*:

*X*→

*Z*have the same property is investigated.

### Linearly ordered compact sets and co-Namioka spaces

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 1001–1004

It is proved that for any Baire space $X$, linearly ordered compact $Y$, and separately continuous mapping $f:\, X \times Y \rightarrow \mathbb{R}$, there exists a $G_{\delta}$-set $A \subseteq X$ dense in $X$ and such that $f$ is jointly continuous at every point of the set $A \times Y$, i.e., any linearly ordered compact is a co-Namioka space.

### Functions of the first Baire class with values in metrizable spaces

Karlova O. O., Mykhailyuk V. V.

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 568–572

We show that every mapping of the first functional Lebesgue class that acts from a topological space into a separable metrizable space that is linearly connected and locally linearly connected belongs to the first Baire class. We prove that the uniform limit of functions of the first Baire class $f_n : \; X \rightarrow Y$ belongs to the first Baire class if $X$ is a topological space and $Y$ is a metric space that is linearly connected and locally linearly connected.

### 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.

### One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 94–101

We investigate the existence of a separately continuous function $f :\; X \times Y \rightarrow \mathbb{R}$ with a one-point set of points of discontinuity in the case where the topological spaces $X$ and $Y$ satisfy conditions of compactness type. In particular, for the compact spaces $X$ and $Y$ and the nonizolated points $x_0 \in X$ and $y_0 \in Y$, we show that the separately continuous function $f :\; X \times Y \rightarrow \mathbb{R}$ with the set of points of discontinuity $\{(x_0, y_0)\}$ exists if and only if sequences of nonempty functionally open set exist in $X$ and $Y$ and converge to $x_0$ and $y_0$, respectively.

### Separately Continuous Functions on Products and Their Dependence on ℵ Coordinates

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1357-1369

We investigate necessary and sufficient conditions on topological products *X* = ∏_{s ∈ s} *X* _{ s } and *Y* = ∏_{t ∈ T} *Y* _{ t } for every separately continuous function *f*: *X × Y* → ℝ to be dependent on at most ℵ coordinates with respect to a certain coordinate.

### Construction of Separately Continuous Functions with Given Restriction

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 716-721

We solve the problem of the construction of separately continuous functions on a product of two topological spaces with given restriction. It is shown, in particular, that, for an arbitrary topological space *X* and a function *g*: *X* → **R** of the first Baire class, there exists a separately continuous function *f*: *X* × *X* → **R** such that *f*(*x*, *x*) = *g*(*x*) for every *x* ∈ *X*.

### Characterization of the sets of discontinuity points of separately continuous functions of many variables on the products of metrizable spaces

Maslyuchenko V. K., Mykhailyuk V. V.

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 740–747

We show that a subset of the product of*n* metrizable spaces is the set of discontinuity points of some separately continuous function if and only if this subset can be represented in the form of the union of a sequence of*F* _{σ}-sets each, of which is locally projectively a set of the first category.

### Dependence of separately continuous functions on n coordinates on products of compact sets

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 822–829

We investigate the problem of the dependence of separately continuous functions on n coordinates on products of spaces each of which is a topological product. In the case where *X* and *Y* are products of completely regular countably compact spaces, we establish necessary and sufficient conditions for such a dependence.

### Separately continuous functions on products of compact sets and their dependence on $\mathfrak{n}$ variables

Maslyuchenko V. K., Mykhailyuk V. V.

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 344-350

By using the theorem on the density of the topological product and the generalized theorem on the dependence of a continuous function defined on a product of spaces on countably many coordinates, we show that every separately continuous function defined on a product of two spaces representable as products of compact spaces with density $≤ \mathfrak{n}$ depends on n variables. In the case of metrizable compact sets, we obtain a complete description of the sets of discontinuity points for functions of this sort.