# Volodymyr MykhaylyukChernivtsi National University · Department of mathematical analysis

Volodymyr Mykhaylyuk

## About

137

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5,300

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395

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Citations since 2016

Introduction

Additional affiliations

January 1995 - present

## Publications

Publications (137)

Given a Riesz space E and $$0 < e \in E$$ 0 < e ∈ E , we introduce and study an order continuous orthogonally additive operator which is an $$\varepsilon $$ ε -approximation of the principal lateral band projection $$Q_e$$ Q e (the order discontinuous lattice homomorphism $$Q_e :E \rightarrow E$$ Q e : E → E which assigns to any element $$x \in E$$...

We investigate the problem of existence of a separately continuous function \(f:X\,{\times }\, Y\rightarrow \mathbb {R}\) defined on a product of topological spaces X and Y with a given discontinuity points set of the form \(A\,{\times }\, B\). Using an approach based on the classical Schwartz function we prove the existence of separately continuou...

We analyze relationship between partial metric spaces and several generalized metric spaces. We first establish that the perfectness of an arbitrary partially metric space X is equivalent to each of the following properties: developability, semi-stratifiability, X has a σ-discrete closed network, X is a β-space with T1. Using this fact we obtain th...

The paper contains a systematic study of the lateral partial order \(\sqsubseteq \) in a Riesz space (the relation \(x \sqsubseteq y\) means that x is a fragment of y) with applications to nonlinear analysis of Riesz spaces. We introduce and study lateral fields, lateral ideals, lateral bands and consistent subsets and show the importance of these...

In 1932 Sierpi\'nski proved that every real-valued separately continuous function defined on the plane $\mathbb R^2$ is determined uniquely on any everywhere dense subset of $\mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $\mathbb R^2$, then they are equal at each point of the plane. Piotrowsk...

We deal with the problem of finding sufficient and necessary conditions on a generalized ordered space X under which for every compact space Y and for every separately continuous function f:X×Y→R: (a) f is of the first Baire class, i.e. X is a Moran space; (b) the function f is a pointwise limit of a sequence of continuous functions which is unifor...

We investigate a possibility of extension of \(F_\sigma \)-measurable and Baire-one maps from subspaces of topological spaces when these maps take values in spaces which can be covered by a sequence of metrizable spaces with special properties.

We construct a separately continuous function $e:E\times K\rightarrow \{0,1\}$ on the product of a Baire space $E$ and a compact space $K$ such that no restriction of $e$ to any non-meagre Borel set in $E\times K$ is continuous. The function $e$ has no points of joint continuity, and, hence, it provides a negative solution of Talagrand’s problem in...

We analyze compactness-like properties of sets in partial metric spaces and obtain the equivalence of several forms of the compactness for partial metric spaces. Moreover, we give a negative answer to a question from [8] on the existence of a bounded complete partial metric on a complete partial metric space.

We study the entropy of a subset A of a Hilbert space H, which equals the infimum of radii of compact bricks in H including A. Our main results show that the entropy can arbitrarily increase under some delicate actions. We prove that the image of a set with finite entropy under the action of a compact operator may have infinite entropy, and the uni...

We introduce a functional Lebesgue classification of multivalued mappings and obtain results on the upper and lower Lebesgue classifications of multivalued mappings F: X × Y ⊸ Z for broad classes of spaces X, Y and Z.

We investigate the possibility of extension of fragmented functions from Lindelöf subspaces of completely regular spaces and find necessary and sufficient conditions on a fragmented Baire-one function to be extendable on any completely regular superspace.

We compare possibilities of extension of bounded and unbounded Baire-one functions from subspaces of topological spaces.

We investigate the possibility of extension of fragmented functions from Lindel\"{o}f subspaces of completely regular spaces and find necessary and sufficient conditions on a fragmented Baire-one function to be extendable on any completely regular superspace

We construct a separately continuous function $e:E\times K\to\{0,1\}$ on the product of a Baire space $E$ and a zero-dimensional compact space $K$ such that no restriction of $e$ to any non-meager Borel set in $E\times K$ is continuous. The function $e$ provides a negative solution of Talagrand's problem in \cite{T}.

We investigate the possibility of extension of $F_\sigma$-measurable and Baire-one maps from subspaces of topological spaces when these maps take values in spaces which covers by a sequence of metrizable spaces with special properties

We introduce and study a new modification of the classical weak Choquet game which we call by an o-game. We show that this game coincides with the σ-game in the class of GO-spaces. Moreover, we obtain a wide class non-Namioka Baire spaces and characterize Namioka spaces in terms of topological games in the class of GO-spaces which can be represente...

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

We prove that if $X$ is a paracompact connected space and $Z=\prod_{s\in S}Z_s$ is a product of a family of equiconnected metrizable spaces endowed with the box topology, then for every Baire-one map $g:X\to Z$ there exists a separately continuous map $f:X^2\to Z$ such that $f(x,x)=g(x)$ for all $x\in X$.

We introduce and study the notions of upper Namioka property, upper Namioka space and upper co-Namioka space which are development of the notions of Namioka property, Namioka space and co-Namioka space on the case of compact-valued mappings. We obtain the following results: the class of upper Namioka spaces consists of Baire spaces with everywhere...

Ми вивчаємо топологічні властивості часткових метрик і частково метричних просторів, зокрема, досліджуємо зв'язок між регулярністю частково метричних просторів і різними аспектами неперервності часткової метрики. Для відображень зі значеннями у частково метричних просторах ми одержуємо аналоги теореми про G _δ-тип множини точок неперервності метриз...

We study a class of fragmented maps which contains all barely continuous maps, scatteredly continuous maps and maps which are pointwise discontinuous on each closed set. We prove that for any Hausdorff space X, a metric space Z and a fragmented map \(f:X\rightarrow Z\) there exists a pointwisely convergent to f sequence of continuous functions \(f_...

We answer a question of O. Kalenda and J. Spurn\'{y} and give an example of a completely regular hereditarily Baire space $X$ and a Baire-one function $f:X\to [0,1]$ which can not be extended to a Baire-one function on $\beta X$.

We give a characterization of countable discrete subspace $A$ of a topological space $X$ such that there exists a (linear) continuous mapping $\varphi:C_p^*(A)\to C_p(X)$ with $\varphi(y)|_A=y$ for every $y\in C_p^*(A)$. Using this characterization we answer two questions of A.~Arhangel'skii. Moreover, we introduce the notion of well-covered subset...

We prove the result on Baire classification of mappings f: X × Y → Z which are continuous with respect to the first variable and belongs to a Baire class with respect to the second one, where X is a PP-space, Y is a topological space and Z is a strongly →-metrizable space with additional properties. We show that for any topological space X, special...

We answer two questions from {\it V.Bykov, On Baire class one functions on a product space, Topol. Appl. {199} (2016) 55--62,} and prove that every Baire one function on a subspace of a countable perfectly normal product is the pointwise limit of a sequence of continuous functions, each depending on finitely many coordinates. It is proved also that...

We solve a problem on a construction of a separately continuous mapping with the given diagonal, which is the pointwise limit of a sequence of continuous mappings valued in an equiconnected space. We construct an example of a closed-valued separately continuous mapping $f:[0,1]^2\to \mathbb R$ with an everywhere discontinuous diagonal. The example...

It is solved the problem on construction of separately continuous functions on product of $n$ topological spaces with given restriction. In particular, it is shown that for every topological space $X$ and $n-1$ Baire class function $g:X\to \mathbb R$ there exists a separately continuous function $f:X^n\to\mathbb R$ such that $f(x,x,\dots,x)=g(x)$ f...

It is solved the problem on constructed of separately continuous functions on product of two topological spaces with given restriction. In particular, it is shown that for every topological space $X$ and first Baire class function $g:X\to \bf R$ there exists separately continuous function $f:X\times X \to \bf R$ such that $f(x,x)=g(x)$ for every $x...

The separately continuity topology is considered and some its properties are investigated. With help of these properties a generalization of Sierpinski theorem on determination of real separately continuous function by its values on an arbitrary dense set is obtained.

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

A notion of strongly Baire space is introduced. Its definition is a transfinite development of some equivalent reformulation of the Baire space definition. It is shown that every strongly Baire space is a Namioka space and every $\beta-\sigma$-unfavorable space is a strongly Baire space.

The cross topology $\gamma$ on a product of topological spaces $X$ and $Y$ is the collection of all sets $G\subseteq X\times Y$ such that the intersection of $G$ with every vertical line and every horizontal line is an open subset of either vertical or horizontal line, respectively. For spaces $X$ and $Y$ from a wide class, which includes all space...

We prove the following two results.
1. If $X$ is a completely regular space such that for every topological space
$Y$ each separately continuous function $f:X\times Y\to\mathbb R$ is of the
first Baire class, then every Lindel\"of subspace of $X$ bijectively
continuously maps onto a separable metrizable space.
2. If $X$ is a Baire space, $Y$ is a c...

We prove the following results.
1. If $X$ is a $\alpha$-favourable space, $Y$ is a regular space, in which
every separable closed set is compact, and $f:X\times Y\to\mathbb R$ is a
separately continuous everywhere jointly discontinuous function, then there
exists a subspace $Y_0\subseteq Y$ which is homeomorphic to $\beta\mathbb N$.
2. There exist...

It is investigated the existence of a separately continuous function
$f:X\times Y\to \mathbb R$ with an onepoint set of discontinuity for
topological spaces $X$ and $Y$ which satisfy compactness type conditions. In
particular, it is shown that for compact spaces $X$ and $Y$ and nonizolated
points $x_0\in X$ and $y_0\in Y$ there exists a separately...

It is investigated necessary and sufficient conditions on topological spaces
$X=\prod\limits _{s\in S}X_s$ and $Y=\prod\limits _{t\in T}Y_t$ for the
dependence of every separately continuous functions $f:X\times Y\to \mathbb R$
on at most $\aleph$ coordinates with respect to the first or the second
variable.

We use a new technique of measures on Boolean algebras to investigate narrow operators on vector lattices. First we prove that, under mild assumptions, every finite rank operator is strictly narrow (before it was known that such operators are narrow). Then we show that every order continuous operator from an atomless vector lattice to a purely atom...

It is obtained necessary and sufficient conditions of dependence on $\aleph$
coordinates for functions of several variables, each of which is a product of
metrizable factors. The set of discontinuity points of such functions is
characterized in the case, when each variable is a product of separable
metrizable spaces.

It is shown that for a function $f:\mathbb R^2\to \mathbb R$ which is
measurable with respect to the first variable and upper semicontinuous
quasicontinuous and increasing with respect to the second variable there exists
a Caratheodory's solution $y(x)=y_0+\int\limits_{x_0}^xf(t,y(t))d\mu(t)$ of the
Cauchy problem $y'(x)=f(x,y(x))$ with the initial...

We prove that, for an interval $X\subseteq \mathbb R$ and a normed space $Z$
diagonals of separately absolute continuous mappings $f:X^2\to Z$ are exactly
such mappings \mbox{$g:X\to Z$} that there is a sequence $(g_n)_{n=1}^{\infty}$
of continuous mappings $g_n:X\to Z$ with $\lim\limits_{n\to\infty}g_n(x)=g(x)$
and \mbox{$\sum\limits_{n=1}^{\infty...

We prove that a function $f(x,y)$ of real variables defined on a rectangle,
having square integrable partial derivatives $f"_{xx}$ and $f"_{yy}$, has
almost everywhere mixed derivatives $f"_{xy}$ and $f"_{yx}$.

We introduce and study adhesive spaces. Using this concept we obtain a
characterization of stable Baire maps $f:X\to Y$ of the class $\alpha$ for wide
classes of topological spaces. In particular, we prove that for a topological
space $X$ and a contractible space $Y$ a map $f:X\to Y$ belongs to the $n$'th
stable Baire class if and only if there exi...

We study the maps between topological spaces whose composition with Baire
class $\alpha$ maps also belongs to the $\alpha$'th Baire class and give
characterizations of such maps

We comment on a Mazur problem from "Scottish Book" concerning second partial
derivatives. It is proved that, if a function $f(x,y)$ of real variables
defined on a rectangle has continuous derivative with respect to $y$ and for
almost all $y$ the function $\,F_y(x):=f'_y(x,y)$ has finite variation, then
almost everywhere on the rectangle there exist...

We prove that, for an interval
X
⊆ ℝ and a normed space
Z
, diagonals of separately absolutely continuous mappings
f
:
X
2
→
Z
are exactly mappings
g : X
→
Z
, which are the sums of absolutely convergent series of continuous functions.

We prove that every continuous function $f:E\to Y$ depends on countably many
coordinates, if $E$ is an $(\aleph_1,\aleph_0)$-invariant
pseudo-$\aleph_1$-compact subspace of a product of topological spaces and $Y$
is a space with a regular $G_\delta$-diagonal. Using this fact for any
$\alpha<\omega_1$ we construct an $(\alpha+1)$-embedded subspace o...

We study properties of strongly separately continuous mappings defined on
subsets of products of topological spaces equipped with the topology of
pointwise convergence. In particular, we give a necessary and sufficient
condition for a strongly separately continuous mapping to be continuous on a
product of an arbitrary family of topological spaces....

In the first part of the paper we prove that for 2 < p, r < infinity every operator T : L-p -> l(r) is narrow. This completes the list of sequence and function Lebesgue spaces X with the property that every operator T : L-p -> X is narrow. Next, using similar methods we prove that every l(2)-strictly singular operator from L-p, 1 < p < infinity, to...

A function f: X → Y between topological spaces is said to be a weakly Gibson function if
$f(\bar G) \subseteq \overline {f(G)} $
for any open connected set G ⊆ X. We call a function f: X → Y segmentary connected if X is topological vector space and f([a, b]) is connected for every segment [a, b] ⊆ X. We show that if X is a hereditarily Baire spac...

A function $f:X\to Y$ between topological spaces is said to be a {\it weakly
Gibson function} if $f(\overline{U})\subseteq \overline{f(U)}$ for any open
connected set \mbox{$U\subseteq X$}. We prove that if $X$ is a locally
connected hereditarily Baire space and $Y$ is a $T_1$-space then an
$F_\sigma$-measurable mapping $f:X\to Y$ is weakly Gibson...

We investigate the Baire classification of mappings f: X × Y → Z, where X belongs to a wide class of spaces which includes all metrizable spaces, Y is a topological space, Z is an equiconnected space, which are continuous in the first variable. We show that for a dense set in X these mappings are functions of a Baire class α in the second variable.

Our main technical tool is a principally new property of compact narrow
operators which works for a domain space without an absolutely continuous norm.
It is proved that for every K\"{o}the $F$-space $X$ and for every locally
convex $F$-space $Y$ the sum $T_1+T_2$ of a narrow operator $T_1:X\to Y$ and a
compact narrow operator $T_2:X\to Y$ is a nar...

It is proved that, for a metric space X and a normed space Z, the diagonals of pointwise Lipschitz mappings f : X 2 → Z are exactly stable pointwise limits of pointwise Lipschitz mappings. The joint Lipschitz property of separately pointwise Lipschitz mappings f : X × Y → Z, where X, Y, and Z are metric spaces, is investigated.

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 [TEX equation: {{\mathb...

We find necessary and sufficient conditions on a Köthe Banach space EE on [0,1][0,1] and a Banach space XX under which a sum of two narrow operators from EE to XX is narrow. Using this condition, we prove that, given a Köthe Banach space EE on [0,1][0,1], there exist a Banach space XX and narrow operators T1,T2:E→XT1,T2:E→X with non-narrow sum T=T1...

It is obtained a general solution of first-order linear partial differential equations in the class of separately differentiable functions.

We prove that for a topological space X, an equiconnected space Z and a Baire-one mapping g:X→Zg:X→Z there exists a separately continuous mapping f:X2→Zf:X2→Z with the diagonal g, i.e. g(x)=f(x,x)g(x)=f(x,x) for every x∈Xx∈X. Under a mild assumptions on X and Z we obtain that diagonals of separately continuous mappings f:X2→Zf:X2→Z are exactly Bair...

A function f:X→Y between topological spaces is said to be a weakly Gibson function if f(U ¯)⊆f(U) ¯ for any open connected set U⊆X. We prove that if X is a locally connected hereditarily Baire space and Y is a T 1 -space then an F σ -measurable mapping f:X→Y is weakly Gibson if and only if for any connected set C⊆X with a dense connected interior t...

It is shown that for any Baire space $X$, linearly ordered compact spaces $Y_1,\dots, Y_n$, compact space $Y\subseteq Y_1\times\cdots \times Y_n$ such that for every parallelepiped $W\subseteq Y_1\times\cdots \times Y_n$ the set $Y\cap W$ is connected, and separately continuous mapping $f:X\times Y\to\mathbb R$ there exists a dense in $X$ $G_\delta...

In the first part of the paper we prove that for $2 < p, r < \infty$ every
operator $T: L_p \to \ell_r$ is narrow. This completes the list of sequence and
function Lebesgue spaces $X$ with the property that every operator $T:L_p \to
X$ is narrow.
Next, using similar methods we prove that every $\ell_2$-strictly singular
operator from $L_p$, $1<p<\i...

A property of boolean independent sequences of pairs of sets is obtained. This completes the proof of Bourgain-Rosenthal's theorem on narrow operators.

The first part of the paper is inspired by a theorem of H. Rosenthal, that if
an operator on $L_1[0,1]$ satisfies the assumption that for each measurable set
$A \subseteq [0,1]$ the restriction $T \bigl|_{L_1(A)}$ is not an isomorphic
embedding, then the operator is narrow.
(Here $L_1(A) = \bigl\{x \in L_1: \,\, {\rm supp} \, x \subseteq A \bigr\}$...

We prove that if Köthe F-spaces X and Y on finite atomless measure spaces (Ω
X
; Σ
X
, µ
X
) and (Ω
Y
; Σ
Y
; µ
Y
), respectively, with absolute continuous norms are isomorphic and have the property
limm(A) ® 0 || m(A) - 1 1A || = 0\mathop {\lim }\limits_{\mu (A) \to 0} \left\| {\mu (A)^{ - 1} 1_A } \right\| = 0
(for µ = µ
X
and µ = µ
Y
, respec...

We introduce a notion of almost antiproximinality of sets in the space L_1 which is a weakening of the notion of antiproximinality. Also we investigate properties of almost antiproximinal sets and establish a method of construction of almost antiproximinal sets.

We prove that for every separately twice differentiable solution $f$ of the
PDE $f"_{xx}=f"_{yy}$ is of the form $f(x,y)=\phi(x+y)+\psi(x-y)$ for some
twice differentiable functions $\phi, \psi$.

A negative solution of Problem 188 posed by Max Eidelheit in {\it the
Scottish Book} concerning superpositions of separately absolutely continuous
functions is presented. We discuss here his and some related problems which
have also negative solutions. Finally, we give an explanation of such negative
answers from the "embeddings of Banach spaces" p...

We establish an asymptotic behavior of the constants in Khintchine's inequality for independent random variables of mean zero.

For a non-isolated point $x$ of a topological space $X$ the network character
$nw_\chi(x)$ is the smallest cardinality of a family of infinite subsets of $X$
such that each neighborhood $O(x)$ of $x$ contains a set from the family. We
prove that (1) each infinite compact Hausdorff space $X$ contains a
non-isolated point $x$ with $nw_\chi(x)=\aleph_...

We introduce the notion of categorical cliquish mapping and show that, for each K
h
C-mapping f: X × Y → Z, where X is a topological space, Y is a space with the first axiom of countability, and Z is a Moore space, with categorical-cliquish horizontal y-sections f
y
, the sets C
y
(f) are residual G
δ-type sets in X for every y ∈ Y.

It is known that if a rearrangement invariant function space E on [0,1] has an unconditional basis then each linear continuous operator on E is a sum of two narrow operators. On the other hand, the sum of two narrow operators in L1 is narrow. To find a general approach to these results, we extend the notion of a narrow operator to the case when the...

Let X and Y be topological spaces such that an arbitrary mapping f: X → Y for which every preimage f
−1 (G) of a set G open in Y is an F
σ-set in X can be represented in the form of the pointwise limit of continuous mappings f
n
: X → Y. We study the problem of subspaces Z of the space Y for which the mappings f: X → Z possess the same property.

It is proved that a differentiable with respect to each variable function
$f:\mathbb R^2\to\mathbb R$ is a solution of the equation $ \frac{\partial
u}{\partial x} + \frac{\partial u}{\partial y}=0$ if and only if there exists a
function $\varphi:\mathbb R\to\mathbb R$ such that $f(x,y)=\varphi(x-y)$. This
gives a positive answer to a question of R...

We investigate the problem of the existence of a noncompact operator T : X-0 subset of X -> Y in terms of the asymptotic structure of separable Banach spaces X and Y. More precisely, for xi = < x(i)>(n)(1) is an element of {X}(n) and eta = < y(i)>(n)(1) is an element of {Y}(n), let T-xi,T-eta be the linear map which sends each x(i) to y(i). We prov...

A connection between the separability and the countable chain condition of spaces with L-property (a topological space X has L-property if for every topological space Y, separately continuous function f:X×Y→ℝ and open set I⊆ℝ, the set f−1(I) is an Fσ-set) is studied. We show that every completely
regular Baire space with the L-property and the coun...

We prove that, for an arbitrary Baire space X, a linearly ordered compact set Y, and a separately continuous mapping ƒ: X × Y → R, there exists a G
δ-set A ⊆ X dense in X and such that the function ƒ is jointly continuous at every point of the set A × Y, i.e., any linearly ordered compact set is a co-Namioka space.

We prove that every point-finite family of nonempty functionally open sets in
a topological space $X$ has the cardinality at most an infinite cardinal
$\kappa$ if and only if $w(X)\leq\kappa$ for every Valdivia compact space
$Y\subseteq C_p(X)$. Correspondingly a Valdivia compact space $Y$ has the
weight at most an infinite cardinal $\kappa$ if and...

We prove that the problem of the existence of a discontinuous separately continuous function f : X×Y → ℝ for any non-discrete Tychonov spaces X, Y of countable pseudocharacter is equivalent to NCPF (Near Coherence of P-filters) which is independent of ZFC. Also for every non-discrete Tychonov space X we find an abelian topological group G of counta...

We solve the problem of constructing separately continuous functions on the product of compact spaces with a given set of discontinuity points. We obtain the following results. 1. For arbitrary ˇ Cech complete spaces X, Y , and a separable compact perfect projectively nowhere dense zero set E X ◊ Y there exists a separately continuous function f :...

We study some properties of the space (L1,X) of all continuous linear operators acting from L1 to a Banach space X. It is proved that every operator T ∈ (L1, X) ``almost'' attains its norm at the entire positive cone of functions supported at some suitable measurable subset , µ(A) > 0. Using this fact and a new elementary technique we prove that ev...

A topological space $Y$ is called a Kempisty space if for any Baire space $X$
every function $f:X\times Y\to\mathbb R$, which is quasi-continuous in the
first variable and continuous in the second variable has the Namioka property.
Properties of compact Kempisty spaces are studied in this paper. In particular,
it is shown that any Valdivia compact...

We introduce a class of $\beta-v$-unfavorable spaces, which contains some
known classes of $\beta$-unfavorable spaces for topological games of Choquet
type. It is proved that every $\beta-v$-unfavorable space $X$ is a Namioka
space, that is for any compact space $Y$ and any separately continuous function
$f:X\times Y\to \mathbb R$ there exists a de...

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 ƒ
n: X → Y belongs to the first Baire class if X is a...

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 se...

A notion of strongly Baire space is introduced. Its definition is a transfinite development of some equivalent reformulation of the Baire space definition. It is shown that every strongly Baire space is a Namioka space and every β − σ-unfavorable space is a strongly Baire space.

We show that the classes B 1 (X,Y) and H 1 (X,Y) of the Baire one mappings and Lebesgue one mappings coincide when X is a hereditarily Baire separable metrizable space and Y is the strict inductive limit of an increasing sequence of metrizable locally convex spaces.