# Karlova O. O.

### 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.

### 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.

### Separately continuous mappings with values in nonlocally convex spaces

Karlova O. O., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1639–1646

We prove that the collection $(X, Y, Z)$ is the Lebesgue triple if $X$ is a metrizable space, $Y$ is a perfectly normal space, and $Z$ is a strongly $\sigma$-metrizable topological vector space with stratification $(Z_m)^{\infty}_{m=1}$, where, for every $m \in \mathbb{N}$, $Z_m$ is a closed metrizable separable subspace of $Z$ arcwise connected and locally arcwise connected.

### 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.