# Functional approach to the normality of mappings

M. Yu. Liseev

###### Abstract

In the article a technique of the usage of $f$-continuous functions (on
mappings) and their families is developed. A proof of the Urysohn’s Lemma for
mappings is presented and a variant of the Brouwer-Tietze-Urysohn Extension
Theorem for mappings is proven. Characterizations of the normality properties
of mappings are given and the notion of a perfect normality of a mapping is
introduced. It seems to be the most optimal in this approach.

Bibliography: 6 names.

Keywords: fiberwise general topology, $f$-continuous mapping,
($\sigma$-)normal mapping, perfectly normal mapping, Urysohn’s Lemma, Brouwer-
Tietze-Urysohn’s Theorem, Vedenisov’s conditions of perfect normality.

00footnotetext: «The paper was published with the financial support of the
Ministry of Education and Science of the Russian Federation as part of the
program of the Moscow Center for Fundamental and Applied Mathematics under the
agreement №075-15-2019-1621.

##  1 Introduction and preliminary information

The fiberwise general topology (also called the topology of continuous maps)
is developing on the basis of general and algebraic topologies. The idea of
extending topological properties of spaces to mappings ("from spaces — to
mappings") was formulated by B. A. Pasinkov in [1] and inspired numerous
studies. The article provides a functional approach to the solution of the
problem of extending normality properties to mappings which was proposed to
the author by B. A. Pasynkov. In particular, the extension to mappings of the
Brouwer-Tietze-Urysohn Theorem about the extension of functions from closed
subsets of normal spaces onto the whole space is considered. The concept of a
perfectly normal mapping is introduced, for which the analogue of Vedenisov’s
condition for perfectly normal spaces is fulfilled.

Definition of a normal mapping [2] has led to the natural definition of a
$co$-perfectly normal mapping (a normal mapping, every open submapping of
which is a $F_{\sigma}$-submapping). In [3], examples of $co$-perfectly normal
mappings that are not hereditarily normal are given, the question of
inheriting normality of a mapping by $F_{\sigma}$-submappings remains open.
The "enhanced" property of the mapping normality — $\sigma$–normality was also
introduced in [3]. Every $\sigma$–normal mapping is normal, and for a constant
$T_{1}$–mapping its normality, $\sigma$–normality and normality of a total
space are equivalent properties [3]. As it turned out, the mapping
$\sigma$–normality is inherited by its $F_{\sigma}$-submappings. Thus, this
approach made it possible to introduce in [3] the concept of a
$co$-$\sigma$–perfect normality of a mapping (i.e. a $\sigma$–normal mapping,
every open subset of which has the type $F_{\sigma}$) and prove that, defined
in this way, the perfect normality of a mapping implies its hereditary
normality, and is a hereditary property.

The concept of a $f$–continuous function on a mapping $f:X\to Y$ was first
introduced by A. Yu. Zubov [4, Theorem 3.1]. Using this concept, he proved an
analogue of Uryson’s Lemma (formulated in [4, Theorem 3.1]) for mappings. It
seems that no further progress in the functional approach to study mappings
(using $f$-continuous functions) occurred.

The designations and the main notions used in the paper are given in § 1. In §
2 a technique of the usage of $f$–continuous functions is developed, and
explicit methods for constructing such functions are given (Lemmas 2.1, 2.4
and Proposition 2.3).

The main results are contained in §§ 3 — 5. Theorem 3.2 of § 3 gives
characterizations of a mapping normality using analogues of the Urysohn’s
Lemmas and the Brouwer–Tietze–Urysohn Extension Theorem for mappings. In its
proof a $f$-continuous function separating disjunct closed subsets is
explicitly constructed. In the given construction a $f$–continuous function
which is continuous on the "limit fiber" of the mapping, is obtaned by
approximating it by stepwise functions. The condition binding the
characterization of normality is a generalized analogue of small Urysohn’s
Lemma (Lemma 3.1).

In § 4 the concept of a family of functions $f$-equicontinuous at a point is
introduced (Definition 4.1). This concept made it possible to characterize the
$\sigma$-normality of a mapping (Theorem 4.3).

In § 5 the concept of a mapping perfect normality is introduced. It uses the
family of functions $f$-equicontinuous at a point and extends Vedenisov’s
perfect normality condition for spaces to mappings (Definition 5.1). The
introduced concept of perfect normality is a hereditary property (Proposition
5.1) and a perfectly normal mapping is hereditarily normal (Theorem 5.3).
Proposition 5.7 shows that this definition of a mapping perfect normality is,
in fact, intermediate, between the previously introduced concepts of
$co$-$\sigma$–perfect normality and $co$–perfect normality, and appears to be
optimal.

All spaces are topological spaces, $\tau_{X}$ is the topology of the space
$X$, $\mathcal{N}(x)$ is the family of open neighborhoods of a point $x\in X$,
mapping $f:X\to Y$ is continuous. The first countable ordinal is
$\omega_{0}=\\{0\\}\cup\mathbb{N}$. The sequence of non-negative integers
$a,a+1,\ldots,b-1,b$ is denoted by $\overline{a,b}$.

Real valued mappings (not necessarily continuous) $\varphi:X\to\mathbb{R}$ are
called functions. The norm of a bounded function $\varphi:X\to\mathbb{R}$ is
$||\varphi||=\sup\limits_{x\in X}|\varphi(x)|$, its oscillation at a point
$x\in X$ is

$\mathrm{osc}_{\varphi}(x)=\inf\\{\sup\limits_{y\in
U}|\varphi(x)-\varphi(y)|:U\in\mathcal{N}(x)\\}\geqslant 0.$

For a non-empty subset $A\subset X$ we set
$\mathrm{osc}_{\varphi}(A)=\mathrm{sup}\\{\mathrm{osc}_{\varphi}(x):x\in A\\}$
and $\mathrm{osc}(\varnothing)=0$.

######  Remark 1.1.

It is easy to check the following.

$\mathrm{osc}_{\varphi}(x)\leqslant 2\cdot||\varphi||$ for any point $x\in X$,
and the function $\varphi$ is continuous at $x$ iff
$\mathrm{osc}_{\varphi}(x)=0$.

For any bounded functions $\varphi:X\to\mathbb{R}$, $\psi:X\to\mathbb{R}$, and
any numbers $\alpha,\beta\in\mathbb{R}$ the following inequality holds

$\mathrm{osc}_{\alpha\varphi+\beta\psi}(A)\leqslant|\alpha|\cdot\mathrm{osc}_{\varphi}(A)+|\beta|\cdot\mathrm{osc}_{\psi}(A).$

######  Lemma 1.1.

For a bounded function $\varphi:X\to\mathbb{R}$ and numbers $a<b$ such that
$b-a>\mathrm{osc}_{\varphi}(X)$ the following hold

$\displaystyle\\{x\in X\ |\ \varphi(x)\leqslant a\\}\cap\mathrm{cl}\\{x\in X\
|\ \varphi(x)\geqslant b\\}=\varnothing,\ \ \mathrm{cl}\\{x\in X\ |\
\varphi(x)\leqslant a\\}\cap\\{x\in X\ |\ \varphi(x)\geqslant
b\\}=\varnothing.$ (1)

###### Proof.

Let the sets $A=\\{x\in X\ |\ \varphi(x)\leqslant a\\}$ and $B=\\{x\in X\ |\
\varphi(x)\geqslant b\\}$ be non-empty.

For any point $x\in A$ there is its open neighborhood $\mathcal{O}x$ such that

$|\varphi(x)-\varphi(z)|<b-a\text{ for any point }z\in\mathcal{O}x.$

Then $A\subset\bigcup\\{\mathcal{O}x\ |\ \varphi(x)\leqslant a\\}=\mathcal{O}$
and $\varphi(z)<b$ for any point $z\in\mathcal{O}$. The set $\mathcal{O}$ is
open and $\mathcal{O}\cap B=\varnothing$. Thus, the first formula in (1) is
proved. The second formula in (1) can be proved similarly. ∎

##  2 $f$–continuity of a function at a point

######  Definition 2.1.

[4, p. 111, the definition of a $\mathcal{G}_{y}(f)$-finally continuous
function] For a mapping $f:X\to Y$, a bounded function
$\varphi:X\to\mathbb{R}$ is called $f$–continuous at a point $y\in Y$

if for any $\varepsilon>0$ there is a neighborhood $\mathcal{O}y$ of the point
$y$ such that $\mathrm{osc}_{\varphi}(f^{-1}\mathcal{O}y)<\varepsilon$.

######  Remark 2.1.

From the Definition 2.1 it follows that a $f$–continuous at a point $y$
function $\varphi$ is continuous at every point $x\in f^{-1}G$ of some
$G_{\delta}$–subset $G\subset Y$, $y\in G$. In particular, at every point of
the fiber $f^{-1}y$.

Any bounded function $\varphi:X\to\mathbb{R}$ is $f$-continuous at the point
$y\in Y$ if $y\not\in\mathrm{cl}\big{(}f(X)\big{)}$.

Linear combination $\alpha\varphi+\beta\psi$ of $f$-continuous at the point
$y\in Y$ functions $\varphi:X\to\mathbb{R}$, $\psi:X\to\mathbb{R}$,
$\alpha,\beta\in\mathbb{R}$, is a $f$-continuous at the point $y\in Y$
function. $f$-continuous at a point $y$ functions form a linear subspace of
functions.

######  Lemma 2.1.

Let for a mapping $f:X\to Y$ and a point $y\in Y$ the pairs
$(\mathcal{O}_{n},\varphi_{n})$, where $\mathcal{O}_{n}$ is a neighborhood of
$y$, $\varphi_{n}:X\to[0,1]$, $n\in\omega_{0}$, are such that

  * (a)

$\mathcal{O}_{n+1}\subset\mathcal{O}_{n}$, $n\in\omega_{0}$,
$\mathcal{O}_{0}=Y$;

  * (b)

the sequence
$\\{\mathrm{osc}_{\varphi_{n}}(f^{-1}\mathcal{O}_{n})\\}_{n\in\mathbb{N}}$
converges to $0$;

  * (c)

the number series
$\sum\limits_{n=0}^{\infty}||\varphi_{n+1}|_{f^{-1}\mathcal{O}_{n+1}}-\varphi_{n}|_{f^{-1}\mathcal{O}_{n+1}}||$
converges.

Then the function $\varphi:X\to[0,1]$

$\varphi(x)=\left\\{\begin{array}[]{ll}\varphi_{n}(x),&x\in
f^{-1}\mathcal{O}_{n}\setminus f^{-1}\mathcal{O}_{n+1},n\in\omega_{0},\\\
\lim\limits_{n\to\infty}\varphi_{n}(x),&x\in\bigcap\limits_{n=0}^{\infty}f^{-1}\mathcal{O}_{n}\\\
\end{array}\right.$

is $f$-continuous at $y\in Y$.

###### Proof.

For any $n,m\in\mathbb{N}$ the following inequality holds

$\displaystyle||\varphi_{n+m}|_{f^{-1}\mathcal{O}_{n+m}}-\varphi_{n}|_{f^{-1}\mathcal{O}_{n+m}}||\leqslant||\varphi_{n+m}|_{f^{-1}\mathcal{O}_{n+m}}-\varphi_{n+m-1}|_{f^{-1}\mathcal{O}_{n+m}}||+\ldots$
(2)
$\displaystyle\ldots+||\varphi_{n+1}|_{f^{-1}\mathcal{O}_{n+1}}-\varphi_{n}|_{f^{-1}\mathcal{O}_{n+1}}||.$

It follows from condition (c) and inequality (2) that the functional sequence
$\\{\varphi_{n}|_{f^{-1}(\bigcap\limits_{n=0}^{\infty}\mathcal{O}_{n})}\\}_{n\in\mathbb{N}}$
converges uniformly on $f^{-1}(\bigcap\limits_{n=0}^{\infty}\mathcal{O}_{n})$.
Thus, the function $\varphi:X\to[0,1]$ is well defined.

Let $\varepsilon>0$ be arbitrary. By condition (b) there exists
$N_{1}\in\mathbb{N}$ such that for all $n\geqslant N_{1}$,
$\mathrm{osc}_{\varphi_{n}}(f^{-1}\mathcal{O}_{n})<\frac{\varepsilon}{3}$. By
condition (c), there exists $N_{2}\in\mathbb{N}$ such that
$\sum\limits_{n=N_{2}}^{\infty}||\varphi_{n+1}|_{f^{-1}\mathcal{O}_{n+1}}-\varphi_{n}|_{f^{-1}\mathcal{O}_{n+1}}||<\frac{\varepsilon}{3}$.
Consider $N=\max\\{N_{1},N_{2}\\}$. For any point $x\in f^{-1}\mathcal{O}_{N}$
there exists a neighborhood $U\in\mathcal{N}(x)$, $U\subset
f^{-1}\mathcal{O}_{N}$, such that for any $\widetilde{x}\in U$ the following
holds

$|\varphi_{N}(x)-\varphi_{N}(\widetilde{x})|<\frac{\varepsilon}{3}.$ (3)

Since $\varphi(x)=\varphi_{N+m}(x)$, for $x\in
f^{-1}\mathcal{O}_{N+m}\setminus f^{-1}\mathcal{O}_{N+m+1}$, or
$\varphi(x)=\lim\limits_{n\to\infty}\varphi_{n}(x)$ otherwise (for
$\varphi(\widetilde{x})$ similarly), then from the condition $N\geqslant
N_{2}$ and the inequality (2) it follows that

$|\varphi(x)-\varphi_{N}(x)|\leqslant\frac{\varepsilon}{3}\text{ and
}|\varphi(\widetilde{x})-\varphi_{N}(\widetilde{x})|\leqslant\frac{\varepsilon}{3}$
(4)

Hence for any point $\widetilde{x}\in U$ the following inequality holds

$|\varphi(x)-\varphi(\widetilde{x})|\leqslant|\varphi(x)-\varphi_{N}(x)|+|\varphi_{N}(x)-\varphi_{N}(\widetilde{x})|+|\varphi_{N}(\widetilde{x})-\varphi(\widetilde{x})|.$

Further, from (3) and (4) it follows that

$|\varphi(x)-\varphi(\widetilde{x})|<\frac{\varepsilon}{3}+\frac{\varepsilon}{3}+\frac{\varepsilon}{3}=\varepsilon.$

Therefore,
$\mathrm{osc}_{\varphi}(f^{-1}\mathcal{O}_{N})\leqslant\varepsilon$, and
$f$-continuity of $\varphi$ at $y$ is proved. ∎

######  Definition 2.2.

(a) For $k\in\mathbb{N}$ the partition $U^{0},\ldots,U^{k-1}$ of the space $X$
is called a regular $k$–partition if

  * (1)

the set $\bigcup\limits_{m=0}^{p}U^{m}$ is closed in $X$ for
$p\in\overline{0,k-1}$

(equivalently the set $\bigcup\limits_{m=l}^{k-1}U^{m}$ is open in $X$ for
$l\in\overline{0,k-1}$);

  * (2)

$(\bigcup\limits_{m=0}^{p}U^{m})\cap\mathrm{cl}(\bigcup\limits_{m=p+2}^{k-1}U^{m})=\varnothing,\
p\in\overline{0,k-3}$, $k\geqslant 3$.

######  Lemma 2.2.

For a regular $k$-partition, $k\geqslant 3$, of the space $X$, the following
holds

($\ast$) $\bigcup\limits_{m=0}^{k-2}\mathrm{int}(U^{m}\cup U^{m+1})=X$.

###### Proof.

Let a regular $k$-partition $\\{U^{m}\\}^{k-1}_{m=0}$ of $X$ be given. For
each $i\in\overline{0,k-3}$ let us show that
$U^{i}\subset\mathrm{int}(U^{i}\cup U^{i+1})$. By condition (2) of Definition
2.2 we have $\bigcup\limits_{m=0}^{i}U^{m}\
\cap\mathrm{cl}\bigcup\limits_{m=i+2}^{k-1}U^{m}=\varnothing$. Then,
$U^{i}\subset
X\setminus\mathrm{cl}\bigcup\limits_{m=i+2}^{k-1}U^{m}=\mathrm{int}\Big{(}X\setminus\mathrm{cl}\bigcup\limits_{m=i+2}^{k-1}U^{m}\Big{)}\subset\mathrm{int}\Big{(}X\setminus\bigcup\limits_{m=i+2}^{k-1}U^{m}\Big{)}=\mathrm{int}\Big{(}\bigcup\limits_{m=0}^{i+1}U^{m}\Big{)}$.
Since $U^{i}\subset\bigcup\limits_{m=i}^{k-1}U^{m}$, then from condition (1)
of Definition 2.2 we obtain that

$U^{i}\subset\mathrm{int}\Big{(}\bigcup\limits_{m=0}^{i+1}U^{m}\Big{)}\cap\bigcup\limits_{m=i}^{k-1}U^{m}=\mathrm{int}\Big{(}\bigcup\limits_{m=0}^{i+1}U^{m}\cap\bigcup\limits_{m=i}^{k-1}U^{m}\Big{)}=\mathrm{int}(U^{i}\cup
U^{i+1}).$

By condition (1) of Definition 2.2 the union $U^{k-2}\cup U^{k-1}$ is open and
$U^{k-2},U^{k-1}\subset\mathrm{int}(U^{k-2}\cup U^{k-1})$. Finally, we obtain

$X=\bigcup\limits_{m=0}^{k-1}U^{m}\subset\bigcup\limits_{m=0}^{k-2}\mathrm{int}(U^{m}\cup
U^{m+1}).$

∎

######  Definition 2.3.

For a mapping $f:X\to Y$ and a point $y\in Y$, the sequence
$\\{\mathcal{O}_{n}\\}_{n\in\omega_{0}}$, $\mathcal{O}_{0}=Y$,
$\mathcal{O}_{n+1}\subset\mathcal{O}_{n}$ of neighborhoods of $y$ and families
$\\{U_{n}^{k}|\ k\in\overline{0,2^{n}-1}\\}_{n\in\omega_{0}}$ of regular
$2^{n}$-partitions of subspaces $f^{-1}\mathcal{O}_{n}$, are called a
consistent family of binary partitions of a mapping $f:X\to Y$ at a point
$y\in Y$ if

$U_{n+1}^{2k}\cup U_{n+1}^{2k+1}=U_{n}^{k}\cap f^{-1}\mathcal{O}_{n+1},\
k\in\overline{0,2^{n}-1}.$

######  Proposition 2.3.

Let for a mapping $f:X\to Y$ and a point $y\in Y$ a consistent family
$\\{\mathcal{O}_{n},\\{U_{n}^{k}|\ k\in\overline{0,2^{n}-1}\\}|\
n\in\mathbb{\omega}_{0}\\}$ of binary partitions of a mapping $f:X\to Y$ at
$y\in Y$ is given.

Then, for the family of functions $\varphi_{n}:X\to[0,1]$

$\varphi_{n}(x)=\left\\{\begin{array}[]{ll}0,&x\in X\setminus
f^{-1}\mathcal{O}_{n},\\\ \frac{k}{2^{n}-1},&x\in U^{k}_{n},\
k\in\overline{0,2^{n}-1},\\\ \end{array}\right.$

the pairs $(\mathcal{O}_{n},\varphi_{n}),\ n\in\omega_{0}$, satisfy the
conditions of Lemma 2.1 and define a $f$–continuous at $y\in Y$ function
$\varphi:X\to[0,1]$

$\varphi(x)=\left\\{\begin{array}[]{ll}\varphi_{n}(x),&x\in
f^{-1}\mathcal{O}_{n}\setminus f^{-1}\mathcal{O}_{n+1},\
n\in\mathbb{\omega}_{0},\\\
\lim\limits_{n\to\infty}\varphi_{n}(x),&x\in\bigcap\limits_{n=0}^{\infty}f^{-1}\mathcal{O}_{n}.\\\
\end{array}\right.$

Moreover, ${\rm
osc}_{\varphi}(f^{-1}\mathcal{O}_{n})\leqslant\frac{1}{2^{n}-1}$.

###### Proof.

Let’s check the fulfillment of conditions (b) and (c) of Lemma 2.1

From the condition ($\ast$) of Lemma 2.2 for a regular $2^{n}$-partition
$\\{U^{k}_{n}|\ k\in\overline{0,2^{n}-1}\\}$ of the subspace
$f^{-1}\mathcal{O}_{n}$ we have

$\text{osc}_{\varphi_{n}}(f^{-1}\mathcal{O}_{n})\leqslant\frac{1}{2^{n}-1},\text{
for }n\in\omega_{0}.$

Thus,
$\lim\limits_{n\to\infty}\mathrm{osc}_{\varphi_{n}}(f^{-1}\mathcal{O}_{n})=0$,
and the condition (b) of Lemma 2.1 is satisfied.

For any point $x\in f^{-1}\mathcal{O}_{n+1}$ we have
$|\varphi_{n+1}(x)-\varphi_{n}(x)|=|\frac{k^{\prime}}{2^{n+1}-1}-\frac{k}{2^{n}-1}|$.
Since $k^{\prime}=2k+1$ or $k^{\prime}=2k$ and $k\in\overline{0,2^{n}-1}$,
then

$\begin{split}|\varphi_{n+1}(x)-\varphi_{n}(x)|&\leqslant\bigg{|}\frac{2k+1}{2^{n+1}-1}-\frac{k}{2^{n}-1}\bigg{|}=\bigg{|}\frac{2^{n}-1-k}{(2^{n+1}-1)(2^{n}-1)}\bigg{|}\leqslant\frac{1}{2^{n+1}-1},\\\
|\varphi_{n+1}(x)-\varphi_{n}(x)|&\leqslant\bigg{|}\frac{2k}{2^{n+1}-1}-\frac{k}{2^{n}-1}\bigg{|}=\bigg{|}\frac{k}{(2^{n+1}-1)(2^{n}-1)}\bigg{|}\leqslant\frac{1}{2^{n+1}-1}.\end{split}$
(5)

So from (5)

$\sum\limits_{n=0}^{\infty}||\varphi_{n+1}|_{f^{-1}\mathcal{O}_{n+1}}-\varphi_{n}|_{f^{-1}\mathcal{O}_{n+1}}||\leqslant\sum_{n=0}^{\infty}\frac{1}{2^{n+1}-1},$

the majorizing series converges and, thus, the condition (c) of Lemma 2.1 is
satisfied.

It follows from the definition of $\varphi$ that
$\text{osc}_{\varphi}(f^{-1}\mathcal{O}_{n})\leqslant\text{osc}_{\varphi_{n}}(f^{-1}\mathcal{O}_{n})\leqslant\frac{1}{2^{n}-1}$,
for $n\in\omega_{0}$. ∎

######  Lemma 2.4.

Let a mapping $f:X\to Y$, a point $y\in Y$ and a family of (bounded)
$f$-continuous at $y\in Y$ functions $\varphi_{n}:X\to\mathbb{R}$,
$n\in\mathbb{N}$, be such that the series
$\sum\limits_{n=1}^{\infty}||\varphi_{n}||$ converges. Then the function
$\varphi(x)=\sum\limits_{n=1}^{\infty}\varphi_{n}(x)$ is $f$-continuous at
$y$.

###### Proof.

Since the series $\sum\limits_{n=1}^{\infty}||\varphi_{n}||$ converges, the
function $\varphi(x)$ is well defined.

Consider an arbitrary $\varepsilon>0$. Let $N\in\mathbb{N}$ be such that
$\sum\limits_{n=N+1}^{\infty}||\varphi_{n}||<\frac{\varepsilon}{4}$. The
$f$-continuity of $\varphi_{n}$ yields that there exists a neighborhood
$\mathcal{O}_{n}$ of $y$ such that
$\mathrm{osc}_{\varphi_{n}}(f^{-1}\mathcal{O}_{n})<\frac{\varepsilon}{2N}$,
$n\leqslant N$.

Let $\mathcal{O}=\bigcap\limits_{n=1}^{N}\mathcal{O}_{n}$. Then from Remark
1.1 it follows that

$\displaystyle\mathrm{osc}_{\varphi}(f^{-1}\mathcal{O})\leqslant\sum\limits_{n=1}^{N}\mathrm{osc}_{\varphi_{n}}(f^{-1}\mathcal{O})$
$\displaystyle+\mathrm{osc}_{\sum\limits_{n=N+1}^{\infty}\varphi_{n}}(f^{-1}\mathcal{O})<$
$\displaystyle<\frac{\varepsilon\cdot N}{2N}$
$\displaystyle+2\sum\limits_{n=N+1}^{\infty}||\varphi_{n}||<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon.$

∎

##  3 Characterizations of normal mappings

Subsets $A$, $B$ of a space $X$ are called separated by neighborhoods in a
subspace $X^{\prime}\subset X$ [1, p. 73] if the sets $A\cap X^{\prime}$ and
$B\cap X^{\prime}$ have disjoint neighborhoods in $X^{\prime}$. For a mapping
$f:X\to Y$, sets $A,B\subset X$ are called $f-$separated by neighborhoods [1,
p. 73], if any point $y\in Y$ has a neighborhood $\mathcal{O}y$, in the
preimage $f^{-1}\mathcal{O}y$ of which the sets $A\cap f^{-1}\mathcal{O}y$ and
$B\cap f^{-1}\mathcal{O}y$ are separated by neighborhoods.

######  Definition 3.1.

[1, p. 73]. A mapping $f:X\to Y$ is said to be prenormal if any two disjoint
closed subsets $A$ and $B$ of $X$ are $f$-separated by neighborhoods.

A mapping $f:X\to Y$ is called normal [2, p. 52] if for any
$\mathcal{O}\in\tau_{Y}$ the restriction
$f_{\mathcal{O}}:f^{-1}\mathcal{O}\rightarrow\mathcal{O}$ of $f$ to
$\mathcal{O}$ is prenormal.

The following statement is a convenient generalization of the ‘‘small Urysohn
Lemma’’ for a normal mapping [5, Proposition 5].

######  Lemma 3.1.

Let $f:X\to Y$ be a normal mapping. Then for any $\mathcal{O}\in\tau_{Y}$ and
any pair of disjoint closed in $f^{-1}\mathcal{O}$ subsets $F$ and $T$, for
any point $y\in\mathcal{O}$ there is a consistent family of binary partitions
$\\{\mathcal{O}_{n},\\{U_{n}^{k}|\ k\in\overline{0,2^{n}-1}\\}|\
n\in\mathbb{\omega}_{0}\\}$ of a mapping $f:X\to Y$ at $y$, such that for any
$n\in\mathbb{N}$

  * (a)

$F\cap f^{-1}\mathcal{O}_{n}\subset U_{n}^{0}$, $T\cap
f^{-1}\mathcal{O}_{n}\subset U_{n}^{2^{n}-1}$;

  * (b)

$F\cap\mathrm{cl}_{f^{-1}\mathcal{O}_{n}}(\bigcup\limits_{k=1}^{2^{n}-1}U_{n}^{k})=\varnothing,\
\
\mathrm{cl}_{f^{-1}\mathcal{O}_{n}}(\bigcup\limits_{k=0}^{2^{n}-2}U_{n}^{k})\cap
T=\varnothing$.

###### Proof.

Without loss of generality, assume that $\mathcal{O}=Y$, disjoint sets $F,\ T$
are closed in $X$ and $y\in Y$. By induction we construct a consistent family
of binary partitions of the mapping $f:X\to Y$ at $y$.

Induction base $n=0$, $\mathcal{O}_{0}=Y$, $U_{0}^{0}=X$.

Induction step. Suppose that for $i\leqslant n$ the neighborhoods
$\mathcal{O}_{i}$ of the point $y$, $\mathcal{O}_{i+1}\subset\mathcal{O}_{i}$,
and the families of regular $2^{i}$-partitions
$\\{\\{U^{j}_{i}|j\in\overline{0,2^{i}-1}\\}|\ i\in\overline{0,n}\\}$ of the
subspaces $f^{-1}\mathcal{O}_{i}$, $i\leqslant n$, satisfying conditions (a),
(b) of lemma and condition of Definition 2.3, have been constructed.

From the normality of the mapping $f$ and [5, Proposition 5] the following
hold. Firstly, for a closed in $f^{-1}\mathcal{O}_{n}$ subset
$\mathrm{cl}_{f^{-1}\mathcal{O}_{n}}\Big{(}\bigcup\limits_{j=1}^{2^{n}-1}U_{n}^{j}\Big{)}$
and its neighborhood $f^{-1}\mathcal{O}_{n}\setminus F$ there is such a
neighborhood $\mathcal{O}_{n+1}^{0}\subset\mathcal{O}_{n}$ of $y$ and an open
in $f^{-1}\mathcal{O}_{n+1}^{0}$ subset $\widetilde{V}^{0}$ such that

$\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}^{0}}\big{(}\bigcup\limits_{j=1}^{2^{n}-1}(U_{n}^{j}\cap
f^{-1}\mathcal{O}^{0}_{n+1})\big{)}\subset\widetilde{V}^{0}\subset\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}^{0}}\widetilde{V}^{0}\subset
f^{-1}\mathcal{O}_{n+1}^{0}\setminus F.$ (6)

Secondly, for $p\in\overline{2,2^{n}-1}$, a closed in $f^{-1}\mathcal{O}_{n}$
subset
$\mathrm{cl}_{f^{-1}\mathcal{O}_{n}}\bigcup\limits_{j=p}^{2^{n}-1}U_{n}^{j}$
and its neighborhood
$f^{-1}\mathcal{O}_{n}\setminus\bigcup\limits_{j=0}^{p-2}U_{n}^{j}$ there are
a neighborhood $\mathcal{O}_{n+1}^{p-1}\subset\mathcal{O}_{n}$ of the point
$y$, and an open in $f^{-1}\mathcal{O}_{n+1}^{p-1}$ set $\widetilde{V}^{p-1}$
such that

$\displaystyle\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}^{p-1}}\big{(}\bigcup\limits_{j=p}^{2^{n}-1}(U_{n}^{j}\cap
f^{-1}\mathcal{O}^{p-1}_{n+1})\big{)}\subset\widetilde{V}^{p-1}$
$\displaystyle\subset\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}^{p-1}}\widetilde{V}^{p-1}\subset$
(7) $\displaystyle\subset
f^{-1}\mathcal{O}_{n}\setminus\bigcup\limits_{j=0}^{p-2}(U_{n}^{j}\cap
f^{-1}\mathcal{O}_{n+1}^{p-1}).$

Thirdly, for a closed in $f^{-1}\mathcal{O}_{n}$ subset $T\cap
f^{-1}\mathcal{O}_{n}$ and its neighborhood $U_{n}^{2^{n}-1}$ there are a
neighborhood $\mathcal{O}_{n+1}^{2^{n}-1}\subset\mathcal{O}_{n}$ of point $y$
and an open subset $\widetilde{V}^{2^{n}-1}$ such that

$T\cap f^{-1}\mathcal{O}_{n+1}^{2^{n}-1}\ \
\subset\widetilde{V}^{2^{n}-1}\subset\
\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}^{2^{n}-1}}\widetilde{V}^{2^{n}-1}\subset
U_{n}^{2^{n}-1}\cap f^{-1}\mathcal{O}_{n+1}^{2^{n}-1}.$ (8)

Consider the sets
$\mathcal{O}_{n+1}=\bigcap\limits_{p=0}^{2^{n}-1}\mathcal{O}_{n}^{p}$ ,
$V^{p}=\widetilde{V}^{p}\cap f^{-1}\mathcal{O}_{n+1}$,
$p\in\overline{0,2^{n}-1}$. For the family of sets
$\\{V^{p}\\}_{p=0}^{2^{n}-1}$ one has:

  * (i)

directly from the construction it follows that $V^{p}\supset V^{p+1}$,
$p\in\overline{0,2^{n}-2}$;

  * (ii)

from inclusions (6) and the fulfillment of condition (a) for a regular
$2^{n}$-partition $\\{U^{k}_{n}\\}_{k=0}^{2^{n}-1}$ of the subspace
$f^{-1}\mathcal{O}_{n}$ it follows that

$F\cap\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}}V^{0}=\varnothing,\ \
\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}}(\bigcup\limits_{k=1}^{2^{n}-1}(U_{n}^{k}\cap
f^{-1}\mathcal{O}_{n+1})\big{)}\subset V^{0};$

  * (iii)

from inclusions (7) it follows for $p\in\overline{0,2^{n}-3}$ that

$(\bigcup\limits_{k=0}^{p}(U_{n}^{k}\cap
f^{-1}\mathcal{O}_{n+1})\big{)}\cap\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}}V^{p+1}=\varnothing,$
$\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}}(\bigcup\limits_{k=p+2}^{2^{n}-1}(U_{n}^{k}\cap
f^{-1}\mathcal{O}_{n+1})\big{)}\subset V^{p+1};$

  * (iv)

from inclusions (8) and the fulfillment of condition (b) for a regular
$2^{n}$-partition$\\{U^{k}_{n}\\}_{k=0}^{2^{n}-1}$ of the subset
$f^{-1}\mathcal{O}_{n}$ it follows that

$(\bigcup\limits_{k=0}^{2^{n}-2}(U_{n}^{k}\cap
f^{-1}\mathcal{O}_{n+1})\big{)}\cap\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}}V^{2^{n}-1}=\varnothing\text{
and }T\cap f^{-1}\mathcal{O}_{n+1}\subset V^{2^{n}-1}.$

Let $U_{n+1}^{2k}=(U_{n}^{k}\setminus V^{k})\cap f^{-1}\mathcal{O}_{n+1}$ ,
$U_{n+1}^{2k+1}=(U_{n}^{k}\cap V^{k})\cap f^{-1}\mathcal{O}_{n+1}$,
$k\in\overline{0,2^{n}-1}$. Then from (ii) – (iv) it follows that
$\\{U^{k}_{n+1}|\ k\in\overline{0,2^{n+1}-1}\\}$ is a regular
$2^{n+1}$-partition of the subspace $f^{-1}\mathcal{O}_{n+1}$ for which the
condition of Definition 2.3 is satisfied.

The constructed sequence of neighborhoods $\mathcal{O}_{n}$ and families of
regular $2^{n}$-partitions of the subspace $f^{-1}\mathcal{O}_{n}$,
$n\in\omega_{0}$ are the consistent family $\\{\mathcal{O}_{n},\\{U_{n}^{k}|\
k\in\overline{0,2^{n}-1}\\}|\ n\in\mathbb{\omega}_{0}\\}$ of binary partitions
of the mapping $f:X\to Y$ at $y$, for which the fulfillment properties (a) and
(b) follows from (ii) and (iv). ∎

######  Theorem 3.2.

For a mapping $f:X\to Y$ the following conditions are equivalent.

  1. (A)

The mapping $f$ is normal;

  2. (B)

For any $\mathcal{O}\in\tau_{Y}$ and any pair of disjoint closed in
$f^{-1}\mathcal{O}$ subsets $F$ and $T$, for any point $y\in\mathcal{O}$ there
is a consistent family of binary partitions $\\{\mathcal{O}_{n},\\{U_{n}^{k}|\
k\in\overline{0,2^{n}-1}\\}|\ n\in\mathbb{\omega}_{0}\\}$ of the mapping
$f:X\to Y$ at $y$, such that for any $n\in\mathbb{N}$

     * (a)

$F\cap f^{-1}\mathcal{O}_{n}\subset U_{n}^{0}$, $T\cap
f^{-1}\mathcal{O}_{n}\subset U_{n}^{2^{n}-1}$;

     * (b)

$F\cap\mathrm{cl}_{f^{-1}\mathcal{O}_{n}}(\bigcup\limits_{k=1}^{2^{n}-1}U_{n}^{k})=\varnothing,\
\
\mathrm{cl}_{f^{-1}\mathcal{O}_{n}}(\bigcup\limits_{k=0}^{2^{n}-2}U_{n}^{k})\cap
T=\varnothing$.

  3. (C)

For any $\mathcal{O}\in\tau_{Y}$ and any pair of disjoint closed in
$f^{-1}\mathcal{O}$ subsets $F$ and $T$, for any point $y\in\mathcal{O}$ there
exist a $f$-continuous at $y$ function $\varphi:X\to[0,1]$ and a neighborhood
$\mathcal{O}y$ of $y$ such that
$\mathrm{osc}_{\varphi}(f^{-1}\mathcal{O}y)<\frac{1}{2}$ and

$F\cap f^{-1}\mathcal{O}y\subset\varphi^{-1}(0)\cap f^{-1}\mathcal{O}y$,
$T\cap f^{-1}\mathcal{O}y\subset\varphi^{-1}(1)\cap f^{-1}\mathcal{O}y$;

$F\cap f^{-1}\mathcal{O}y\subset
f^{-1}\mathcal{O}y\setminus\mathrm{cl}_{f^{-1}\mathcal{O}y}(\varphi^{-1}[\frac{1}{2},1]\cap
f^{-1}\mathcal{O}y)$, $T\cap
f^{-1}\mathcal{O}y\subset\mathrm{int}_{f^{-1}\mathcal{O}y}(\varphi^{-1}[\frac{1}{2},1]\cap
f^{-1}\mathcal{O}y)$.

  4. (D)

Let $\mathcal{O}\in\tau_{Y}$, $F$ is a non-empty closed subset of
$f^{-1}\mathcal{O}$ and $y\in\mathcal{O}$ is arbitrary. Then, for any
$\widetilde{f}$-continuous at $y$ function
$\widetilde{\varphi}:F\to\mathbb{R}$ of the mapping
$\widetilde{f}:F\to\mathcal{O}$, ($\widetilde{f}(x)=f(x)$ for $x\in F$) there
is a $f$-continuous at $y$ function $\varphi:X\to\mathbb{R}$ for the mapping
$f:X\to Y$ such that

     * (a)

$\widetilde{\varphi}|_{F\cap f^{-1}G}=\varphi|_{F\cap f^{-1}G}$, where $G$ is
a $G_{\delta}$–subset of $\mathcal{O}$ and $y\in G$
(in particular, $\widetilde{\varphi}|_{F\cap f^{-1}y}=\varphi|_{F\cap
f^{-1}y}$) ,

     * (b)

$||\varphi||\leqslant||\widetilde{\varphi}||$,

     * (c)

for any $\varepsilon>0$ there is a neighborhood
$\mathcal{O}(\varepsilon)\subset\mathcal{O}$ of $y$ such that
$||\widetilde{\varphi}|_{F\cap f^{-1}\mathcal{O}(\varepsilon)}-\varphi|_{F\cap
f^{-1}\mathcal{O}(\varepsilon)}||<\varepsilon$.

###### Proof.

(A) $\Rightarrow$ (B) by Lemma 3.1.

(B) $\Rightarrow$ (C). By (B) for a pair of disjoint closed in
$f^{-1}\mathcal{O}$ subsets $F$ and $T$ there exists a consistent family of
binary partitions $\\{\mathcal{O}_{n},\\{U^{k}_{n}|\
k\in\overline{0,2^{n}-1}\\}|\ n\in\omega_{0}\\}$ of the mapping $f:X\to Y$ at
$y$ for which conditions (a) and (b) hold.

In Proposition 2.3 the $f$-continuous at $y\in Y$ function

$\varphi(x)=\left\\{\begin{array}[]{ll}\varphi_{n}(x),&x\in
f^{-1}\mathcal{O}_{n}\setminus f^{-1}\mathcal{O}_{n+1},\
n\in\mathbb{\omega}_{0},\\\
\lim\limits_{n\to\infty}\varphi_{n}(x),&x\in\bigcap\limits_{n=0}^{\infty}f^{-1}\mathcal{O}_{n}.\\\
\end{array}\right.$

according to a consistent family of binary partitions
$\\{\mathcal{O}_{n},\\{U^{k}_{n}|\ k\in\overline{0,2^{n}-1}\\}|\
n\in\omega_{0}\\}$ of the mapping $f:X\to Y$ at $y$ was constructed, where
$\varphi_{n}:X\to\mathbb{R}$ are the following

$\varphi_{n}(x)=\left\\{\begin{array}[]{ll}0,&x\in X\setminus
f^{-1}\mathcal{O}_{n},\\\ \frac{k}{2^{n}-1},&x\in U^{k}_{n},\
k\in\overline{0,2^{n}-1},\ n\in\omega_{0}.\\\ \end{array}\right.$

In this case, firstly, $\varphi(x)=0$ for $x\in F$ (since $F\subset
U_{n}^{0}$, $n\in\mathbb{N}$), $\varphi(x)=1$ for $x\in T$ (since $T\subset
U_{n}^{2^{n}-1}$, $n\in\mathbb{N}$).

Secondly, due to the $f$-continuity of $\varphi$ there is a neighborhood
$\mathcal{O}y$ of $y$ such that
$\mathrm{osc}_{\varphi}(f^{-1}\mathcal{O}y)<\frac{1}{2}$. Then by Lemma 1.1

$F\cap f^{-1}\mathcal{O}y\subset\varphi^{-1}(0)\cap f^{-1}\mathcal{O}y\subset
f^{-1}\mathcal{O}y\setminus\mathrm{cl}_{f^{-1}\mathcal{O}y}(\varphi^{-1}[\tfrac{1}{2},1]\cap
f^{-1}\mathcal{O}y).$

Thirdly, for any $n\in\mathbb{N}$

$\varphi_{n}^{-1}[\tfrac{1}{2},1]=\bigcup\limits_{k=2^{n-1}}^{2^{n}-1}U_{n}^{k}=U^{1}_{1}\cap
f^{-1}\mathcal{O}_{n}.$

Hence,

$T\cap f^{-1}\mathcal{O}y\subset U^{1}_{1}\cap
f^{-1}\mathcal{O}y\subset\mathrm{int}_{f^{-1}\mathcal{O}y}(\varphi^{-1}[\tfrac{1}{2},1]\cap
f^{-1}\mathcal{O}y).$

(C) $\Rightarrow$ (D). If $Y\neq\mathcal{O}$ and the
$f_{\mathcal{O}}$-continuous at $y$ function
$\varphi:f^{-1}\mathcal{O}\to\mathbb{R}$ (for a restriction
$f_{\mathcal{O}}:f^{-1}\mathcal{O}\to\mathcal{O}$ of the mapping $f:X\to Y$)
which satisfies conditions similar to the conditions of (D) is constructed,
then defining, additionally, the function $\varphi$ by the value 0 at the
points $X\setminus f^{-1}\mathcal{O}$ we obtain the required function.
Therefore, one can assume that $Y=\mathcal{O}$, the subset $F$ is closed in
$X$ and $y\in Y$.

If $y\not\in\mathrm{cl}\big{(}f(X)\big{)}$, then the constant function
$\varphi$ taking the value 0 on $X$ is the required one.

Otherwise, put $\varphi_{0}=\widetilde{\varphi}$, $\mu_{0}=||\varphi_{0}||$.
If $||\varphi_{0}||=0$, then put $\varphi\equiv 0$.

Otherwise, from the $\widetilde{f}$-continuity of the function
$\widetilde{\varphi}$ at $y$, let $\mathcal{O}^{\prime}_{0}$ be a neighborhood
of $y$ such that $\mathrm{osc}_{\widetilde{\varphi}}(F\cap
f^{-1}\mathcal{O}^{\prime}_{0})<\frac{\mu_{0}}{3}$,

$\begin{split}P_{0}=\text{cl}_{f^{-1}\mathcal{O}^{\prime}_{0}}\\{x\in F\cap
f^{-1}\mathcal{O}^{\prime}_{0}\ |\
\varphi_{0}(x)\leqslant-\tfrac{\mu_{0}}{3}\\},\\\
Q_{0}=\text{cl}_{f^{-1}\mathcal{O}^{\prime}_{0}}\\{x\in F\cap
f^{-1}\mathcal{O}^{\prime}_{0}\ |\
\varphi_{0}(x)\geqslant\tfrac{\mu_{0}}{3}\\}.\end{split}$

Then, by Lemma 1.1, the closed in $f^{-1}\mathcal{O}^{\prime}_{0}$ subsets
$P_{0}$ and $Q_{0}$ are disjoint. By (C) (replacing the segment $[0,1]$ with
$[-\frac{\mu_{0}}{3},\frac{\mu_{0}}{3}]$), there exist a $f$-continuous at $y$
function $\psi_{0}:X\to[-\frac{\mu_{0}}{3},\frac{\mu_{0}}{3}]$ and a
neighborhood $\mathcal{O}_{0}\subset\mathcal{O}^{\prime}_{0}$ of the point $y$
such that

$\begin{split}P_{0}\cap
f^{-1}\mathcal{O}_{0}\subset\psi^{-1}_{0}(-\tfrac{\mu_{0}}{3}),\ Q_{0}\cap
f^{-1}\mathcal{O}_{0}\subset\psi^{-1}_{0}(\tfrac{\mu_{0}}{3}),\\\ P_{0}\cap
f^{-1}\mathcal{O}_{0}\subset
f^{-1}\mathcal{O}_{0}\setminus\mathrm{cl}_{f^{-1}\mathcal{O}_{0}}(\psi_{0}^{-1}([0,\tfrac{\mu_{0}}{3}])\cap
f^{-1}\mathcal{O}_{0}),\\\ Q_{0}\cap
f^{-1}\mathcal{O}_{0}\subset\mathrm{int}_{f^{-1}\mathcal{O}_{0}}(\psi_{0}^{-1}([0,\tfrac{\mu_{0}}{3}])\cap
f^{-1}\mathcal{O}_{0}),\\\ \mathrm{osc}_{\psi_{0}}(F\cap
f^{-1}\mathcal{O}_{0})<\tfrac{\mu_{0}}{3}.\end{split}$

Let $\varphi_{1}=\varphi_{0}-\psi_{0}:F\cap
f^{-1}\mathcal{O}_{0}\to\mathbb{R}$. By Remark 2.1 the function $\varphi_{1}$
is $f|_{F\cap f^{-1}\mathcal{O}_{0}}$–continuous at $y$ and
$||\varphi_{1}||=\mu_{1}\leqslant\tfrac{2\mu_{0}}{3}$.

By induction we construct a non-increasing (with respect to embedding)
sequence of neighborhoods $\\{\mathcal{O}_{n}\\}_{n=0}^{\infty}$ of $y$, a
sequence of $f$-continuous at $y$ functions $\psi_{n}:X\to\mathbb{R}$ and a
$\widetilde{f}$-continuous at $y$ functions $\varphi_{n}:F\cap
f^{-1}\mathcal{O}_{n}\to\mathbb{R}$ such that
$\varphi_{n+1}=\varphi_{n}-\psi_{n}$,
$||\psi_{n}||\leqslant\frac{\mu_{n}}{3}$,
$||\varphi_{n+1}||=\mu_{n+1}\leqslant\frac{2\mu_{n}}{3}$.

$||\varphi_{n}||\leqslant\bigg{(}\frac{2}{3}\bigg{)}^{n}\mu_{0},\
||\psi_{n}||\leqslant\bigg{(}\frac{2}{3}\bigg{)}^{n}\frac{\mu_{0}}{3}.\ $ (9)

Let

$\varphi(x)=\sum\limits_{k=0}^{\infty}\psi_{n}(x).$ (10)

From the conditions (9) it follows that the series
$\sum\limits_{k=0}^{\infty}||\psi_{n}||$ converges. By Lemma 2.4 the function
$\varphi:X\to\mathbb{R}$ is $f$-continuous at $y$. The inequality
$||\varphi||\leqslant||\widetilde{\varphi}||$ follows from the second
inequality of (9). Indeed, from (10) it follows that

$||\varphi||\leq\sum\limits_{k=0}^{\infty}||\psi_{n}||\leqslant\frac{\mu_{0}}{3}\sum\limits_{k=0}^{\infty}\bigg{(}\frac{2}{3}\bigg{)}^{k}=||\widetilde{\varphi}||.$

From the equality (10), equation
$\varphi_{n+1}=\varphi_{n}-\psi_{n}=\widetilde{\varphi}-\sum\limits_{k=0}^{n}\psi_{n}$
and the first inequality of (9) one has
$\widetilde{\varphi}|_{F\cap\bigcap\limits_{n=0}^{\infty}f^{-1}\mathcal{O}_{n}}=\varphi|_{F\cap\bigcap\limits_{n=0}^{\infty}f^{-1}\mathcal{O}_{n}}$
(in particular, $\widetilde{\varphi}|_{F\cap f^{-1}y}=\varphi|_{F\cap
f^{-1}y}$) and the last condition of item (D) holds.

(D) $\Rightarrow$ (A). Let $\mathcal{O}\in\tau_{Y}$, $F$ and $T$ are closed
disjoint subsets of $f^{-1}\mathcal{O}$, $y\in\mathcal{O}$.

The function

$\widetilde{\varphi}(x)=\left\\{\begin{array}[]{cl}0,&x\in F,\\\ 1,&x\in T\\\
\end{array}\right.$ (11)

is $\widetilde{f}$-continuous at $y$ for the restriction $\widetilde{f}:F\cup
T\to{\mathcal{O}}$ of the mapping $f:X\to Y$. The set $F\cup T$ is closed in
$f^{-1}\mathcal{O}$.

By condition (D) there is a $f$-continuous at $y$ function $\varphi:X\to[0,1]$
for the mapping $f:X\to Y$ such that $\widetilde{\varphi}|_{(F\cup T)\cap
f^{-1}G}=\varphi|_{(F\cup T)\cap f^{-1}G}$, where $G$ is a $G_{\delta}$-subset
of $\mathcal{O}$ and $y\in G$ (in particular $\widetilde{\varphi}|_{(F\cup
T)\cap f^{-1}y}=\varphi|_{(F\cup T)\cap f^{-1}y}$),
$||\varphi||\leqslant||\widetilde{\varphi}||$, and for
$\varepsilon=\frac{1}{4}$ there is a neighborhood
$\mathcal{O}^{\prime}\subset\mathcal{O}$ of $y$ such that
$||\widetilde{\varphi}|_{(F\cup T)\cap
f^{-1}\mathcal{O}^{\prime}}-\varphi|_{(F\cup T)\cap
f^{-1}\mathcal{O}^{\prime}}||<\frac{1}{4}$ .

Since the function $\varphi:X\to\mathbb{R}$ is $f$-continuous at $y$ for the
mapping $f:X\to Y$, then there exists a neighborhood
$\mathcal{O}y\subset\mathcal{O}^{\prime}$ such that

$\mathrm{osc}_{\varphi}(f^{-1}\mathcal{O}y)<\frac{1}{4}.$

Then, by Lemma 1.1 we have $\big{(}\varphi^{-1}[0,\frac{1}{4}]\cap
f^{-1}\mathcal{O}y\big{)}\cap\big{(}\mathrm{cl}_{f^{-1}\mathcal{O}y}(\varphi^{-1}[\frac{3}{4},1]\cap
f^{-1}\mathcal{O}y)\big{)}=\varnothing$. Open sets
$V=\mathrm{int}_{f^{-1}\mathcal{O}y}\big{(}\varphi^{-1}[0,\frac{1}{4}]\cap
f^{-1}\mathcal{O}y\big{)}$ and
$U=\mathrm{int}_{f^{-1}\mathcal{O}y}\big{(}\varphi^{-1}[\frac{3}{4},1]\cap
f^{-1}\mathcal{O}y\big{)}$ are disjoint. From (11) it follows that $F\cap
f^{-1}\mathcal{O}y\subset V$, $T\cap f^{-1}\mathcal{O}y\subset U$. Thus, the
mapping $f:X\to Y$ is normal. ∎

######  Remark 3.1.

1\. Implication (A) $\Rightarrow$ (C) of Theorem 3.2 is the Urysohn’s Lemma
for mappings, which is formulated in [4].

2\. Statement (D) of Theorem 3.2 is a variant of extension of the Brouwer-
Tietze-Urysohn Theorem to mappings.

3\. The extension of a $f|_{F}$-continuous at $y$ function, is equivalent to
the extension of a $f|_{F}$-continuous at $y$ bounded mapping into a Banach
space.

4\. In the case of a constant mapping $f:X\to\\{y\\}$ the statement of Theorem
3.2 coincides with the statement of Urysohn’s Lemma [6, p. 75, Theorem 1.5.10]
and the Brouwer–Tietze–Urysohn Theorem[6, p. 116, Theorem 2.1.8] for spaces.

##  4 Family of functions $f$-equicontinuous at a point and $\sigma$-normality
of a mapping

######  Definition 4.1.

For a mapping $f:X\to Y$, a family of functions
$\\{\varphi_{n}:X\to[0,1]\\}_{n\in\mathbb{N}}$ is called a $f$-equicontinuous
family of functions at a point $y\in Y$ if for any $\varepsilon>0$ there is a
neighborhood $\mathcal{O}y$ of $y$ such that for any $n\in\mathbb{N}$

$\mathrm{osc}_{\varphi_{n}}(f^{-1}\mathcal{O}y)<\varepsilon.$

######  Remark 4.1.

Each function $\varphi_{n}$ of a $f$–equicontinuous family
$\\{\varphi_{n}\\}_{n\in\mathbb{N}}$ of functions at $y$ is $f$–continuous at
$y$, the converse is not true.

######  Definition 4.2.

[3, Definition 7]. A mapping $f:X\rightarrow Y$ is said to be a
$\sigma$–prenormal if for any $F_{\sigma}$–set
$T=\bigcup\limits_{l=1}^{\infty}T_{l}$, where the subset $T_{l}$ is closed in
$X$, $l\in\mathbb{N}$, and a closed in $X$ subset $F$ such that $T\cap
F=\varnothing$, for any point $y\in Y$ there are its neighborhood
$\mathcal{O}y$ and a family $\\{\mathcal{O}_{l}\\}_{l=1}^{\infty}$ of open in
$f^{-1}\mathcal{O}y$ sets such that $T_{l}\cap
f^{-1}\mathcal{O}y\subset\mathcal{O}_{l}$, $l\in\mathbb{N}$, and
$\Big{(}\bigcup\limits_{l=1}^{\infty}\mathrm{cl}_{f^{-1}\mathcal{O}y}(\mathcal{O}_{l})\Big{)}\cap
F=\varnothing$.

A mapping $f$ is called a $\sigma$–normal if the restriction
$f_{\mathcal{O}}:f^{-1}\mathcal{O}\to\mathcal{O}$ of $f$ to $\mathcal{O}$ is
$\sigma$–prenormal for any $\mathcal{O}\in\tau_{Y}$.

It should be noted that in the case of a constant mapping its prenormality,
normality, $\sigma$–prenormality and $\sigma$–normality are equivalent.
However, unlike normality, $\sigma$–normality of the mapping is inherited by
$F_{\sigma}$–submappings [3, Theorem 6].

######  Lemma 4.1.

A mapping $f:X\to Y$ $\sigma$ is normal iff for any $\mathcal{O}\in\tau_{Y}$,
any point $y\in\mathcal{O}$ and any $F_{\sigma}$-subset
$T=\bigcup\limits_{l=1}^{\infty}T_{l}$ in $f^{-1}\mathcal{O}$, and its
neighborhood $U\subset f^{-1}\mathcal{O}$, there is a neighborhood
$\mathcal{O}y\subset\mathcal{O}$ of the point $y$ such that in
$f^{-1}\mathcal{O}y$ there are neighborhoods $V_{l}$ of the sets $T_{l}\cap
f^{-1}\mathcal{O}y$, $l\in\mathbb{N}$, and

$T_{l}\cap f^{-1}\mathcal{O}y\subset V_{l}\subset{\rm
cl}_{f^{-1}\mathcal{O}y}V_{l}\subset U\cap f^{-1}\mathcal{O}y,\
l\in\mathbb{N}.$

###### Proof.

Necessity.  Let us fix an arbitrary subset $\mathcal{O}\in\tau_{Y}$ and a
point $y\in\mathcal{O}$. For closed in $f^{-1}\mathcal{O}$ sets $T_{l}$,
$l\in\mathbb{N}$, and their neighborhood $U$ consider the set
$f^{-1}\mathcal{O}\setminus U$. It is closed in $f^{-1}\mathcal{O}$ and
$T\cap(f^{-1}\mathcal{O}\setminus U)=\varnothing$. Since the mapping $f$ is
$\sigma$-normal, then for $y$ there are a neighborhood
$\mathcal{O}y\subset\mathcal{O}$ and neighborhoods $V_{l}\subset
f^{-1}\mathcal{O}y$ of subsets $T_{l}\cap f^{-1}\mathcal{O}y$ such that

$\bigcup\limits_{l=1}^{\infty}\mathrm{cl}_{f^{-1}\mathcal{O}y}(V_{l})\cap(f^{-1}\mathcal{O}y\setminus
U)=\varnothing,\ l\in\mathbb{N}.$

Thus, for each $l\in\mathbb{N}$ we have $T_{l}\cap f^{-1}\mathcal{O}y\subset
V_{l}\subset{\rm cl}_{f^{-1}\mathcal{O}y}V_{l}\subset U\cap
f^{-1}\mathcal{O}y$.

Sufficiency. Let $\mathcal{O}\in\tau_{Y}$, $y\in\mathcal{O}$ an arbitrary
fixed point and consider two disjoint subsets $F$ and $T$ of the set
$f^{-1}\mathcal{O}$ such that

$T=\bigcup\limits_{l=1}^{\infty}T_{l}$ is $F_{\sigma}$–subset, and the subsets
$T_{l}$ are closed in $f^{-1}\mathcal{O}$, $l\in\mathbb{N}$,

$F$ is closed in $f^{-1}\mathcal{O}$.

By the assumption of the theorem, for the point $y$, subsets $T_{l}$,
$l\in\mathbb{N}$ and their neighborhood $f^{-1}\mathcal{O}\setminus F$ there
exist a neighborhood $\mathcal{O}y$ of $y$ and open sets $V_{l}$,
$l\in\mathbb{N}$ such that

$T_{l}\cap f^{-1}\mathcal{O}y\subset V_{l}\subset{\rm
cl}_{f^{-1}\mathcal{O}y}V_{l}\subset f^{-1}\mathcal{O}y\setminus F,\
l\in\mathbb{N}.$

Therefore,
$\bigcup\limits_{l=1}^{\infty}\mathrm{cl}_{f^{-1}\mathcal{O}y}(V_{l})\cap
F=\varnothing$. The $\sigma$-prenormality of the mapping
$f_{\mathcal{O}}:f^{-1}\mathcal{O}\rightarrow\mathcal{O}$ is proven and the
mapping $f$ is normal. ∎

In [1] the submapping $f\big{|}_{X_{0}}:X_{0}\to Y$ is defined as the
restriction of the mapping $f:X\to Y$ to a subset $X_{0}\subset X$. The
submapping $f\big{|}_{X_{0}}:X_{0}\to Y$ is called an open (closed) submapping
[1] if $X_{0}$ is an open (closed) subset of $X$. By disjoint submappings
$f|_{A}:A\to Y$, $f|_{B}:B\to Y$ of the mapping $f:X\to Y$ we understand that
the subsets $A$ and $B$ are disjoint.

######  Definition 4.3.

[3, Definition 6]. A submapping $f|_{X_{0}}:X_{0}\to Y$ is said to be of
$F_{\sigma}$-type (or is a $F_{\sigma}$–submapping) if for any point $y\in Y$
there is its neighborhood $\mathcal{O}y\subset Y$ such that
$(f\big{|}_{X_{0}})^{-1}\mathcal{O}y$ is a $F_{\sigma}$–subset of
$f^{-1}\mathcal{O}y$.

A convenient generalization of the “small Urysohn Lemma” for $\sigma$-normal
mappings is the following.

######  Lemma 4.2.

Let $f:X\to Y$ be a $\sigma$–normal mapping. Then for any
$\mathcal{O}\in\tau_{Y}$ and any disjoint closed submapping
$f_{\mathcal{O}}|_{F}:F\to\mathcal{O}$ and $F_{\sigma}$-submapping
$f_{\mathcal{O}}|_{T}:T\to Y$ of the mapping
$f_{\mathcal{O}}:f^{-1}\mathcal{O}\to\mathcal{O}$ and any point
$y\in\mathcal{O}$ there are consistent families of binary partitions
$\Gamma_{l}=\\{\mathcal{O}_{n},\\{U_{n}^{k}(l)|\ k\in\overline{0,2^{n}-1}\\}|\
n\in\omega_{0}\\}$, $l\in\mathbb{N}$, of a mapping $f:X\to Y$ at $y$ such that
for any $n\in\mathbb{N}$

  * (a)

$T\cap f^{-1}\mathcal{O}_{1}=\bigcup\limits_{l=1}^{\infty}T_{l}$ , where
$T_{l}$ is closed in $f^{-1}\mathcal{O}_{1}$, $l\in\mathbb{N}$;

  * (b)

$F\cap f^{-1}\mathcal{O}_{n}\subset U_{n}^{0}(l)$, $T_{l}\cap
f^{-1}\mathcal{O}_{n}\subset U_{n}^{2^{n}-1}(l)$, $l\in\mathbb{N}$;

  * (c)

$F\cap\mathrm{cl}_{f^{-1}\mathcal{O}_{n}}(\bigcup\limits_{k=1}^{2^{n}-1}U_{n}^{k}(l))=\varnothing,\
\
\mathrm{cl}_{f^{-1}\mathcal{O}_{n}}(\bigcup\limits_{k=0}^{2^{n}-2}U_{n}^{k}(l))\cap
T_{l}=\varnothing$, $l\in\mathbb{N}$.

###### Proof.

Without loss of generality, we assume that $\mathcal{O}=Y$, the sets $F,\ T$
are disjoint in $X$ and $y\in Y$. Let $\mathcal{O}_{0}=Y$, $U_{0}^{0}(l)=X$,
$l\in\mathbb{N}$. Let’s construct by induction the consistant families of
binary partitions $\Gamma_{l}$, $l\in\mathbb{N}$, for the mapping $f:X\to Y$
at the point $y$.

Induction base $n=1$. Since $f|_{T}$ is a $F_{\sigma}$–submapping, there
exists a neighborhood $\mathcal{O}^{\prime}_{1}\in\tau_{Y}$ of $y$ such that
$T\cap f^{-1}\mathcal{O}^{\prime}_{1}=\bigcup\limits_{l=1}^{\infty}T_{l}$,
where $T_{l}$ is closed in $f^{-1}\mathcal{O}^{\prime}_{1}$, $l\in\mathbb{N}$.
By Lemma 4.1 there is a neighborhood $\mathcal{O}y\subset\mathcal{O}$ of $y$
such that in $f^{-1}\mathcal{O}y$ there exist neighborhoods $V_{l}$ of the
sets $T_{l}\cap f^{-1}\mathcal{O}y$, $l\in\mathbb{N}$ such that

$T_{l}\cap f^{-1}\mathcal{O}y\subset V_{l}\subset{\rm
cl}_{f^{-1}\mathcal{O}y}V_{l}\subset U\cap f^{-1}\mathcal{O}y,\
l\in\mathbb{N}.$

Let $\mathcal{O}_{1}=\mathcal{O}y$, $U^{0}_{1}=f^{-1}\mathcal{O}_{1}\setminus
V_{l}$, $U^{1}_{1}=f^{-1}\mathcal{O}_{1}\cap V_{l}$, $l\in\mathbb{N}$. Thus,
the validity of condition (a) is established.

Induction step. Suppose that for $i\leqslant n$ and for any $l\in\mathbb{N}$
the neighborhoods $\mathcal{O}_{i}$ of the point $y$,
$\mathcal{O}_{i+1}\subset\mathcal{O}_{i}$, $i<n$, and the families of regular
$2^{i}$-partitions $\\{\\{U^{j}_{i}(l)|j\in\overline{0,2^{i}-1}\\}|\
i\in\overline{0,n}\\}$ of subspaces $f^{-1}\mathcal{O}_{i}$, satisfying
conditions (b), (c) of the lemma and condition of the Definition 2.3 have been
constructed.

From the $\sigma$-normality of the mapping $f$, using Lemma 4.1, we have the
following.

Firstly, for the $F_{\sigma}$–subset
$\bigcup\limits_{l=1}^{\infty}\mathrm{cl}_{f^{-1}\mathcal{O}_{n}}\Big{(}\bigcup\limits_{j=1}^{2^{n}-1}U_{n}^{j}(l)\Big{)}$
of $f^{-1}\mathcal{O}_{n}$ and its neighborhood
$f^{-1}\mathcal{O}_{n}\setminus F$ there are a neighborhood
$\mathcal{O}_{n+1}^{0}\subset\mathcal{O}_{n}$ of $y$ and open in
$f^{-1}\mathcal{O}_{n+1}^{0}$ subsets $\widetilde{V}^{0}(l)$, $l\in\mathbb{N}$
such that

$\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}^{0}}\big{(}\bigcup\limits_{j=1}^{2^{n}-1}(U_{n}^{j}(l)\cap
f^{-1}\mathcal{O}^{0}_{n+1})\big{)}\subset\widetilde{V}_{l}^{0}\subset\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}^{0}}\widetilde{V}_{l}^{0}\subset
f^{-1}\mathcal{O}_{n+1}^{0}\setminus F,\ l\in\mathbb{N}.$ (12)

Secondly, provided that $p\in\overline{2,2^{n}-1}$, for the
$F_{\sigma}$-subset
$\bigcup\limits_{i=1}^{\infty}\big{(}\mathrm{cl}_{f^{-1}\mathcal{O}_{n}}\big{(}\bigcup\limits_{m=p}^{2^{n}-1}U_{n}^{m}(l)\big{)}\big{)}$
of $f^{-1}\mathcal{O}_{n}$, and its neighborhood
$\bigcup\limits_{i=1}^{\infty}\big{(}\bigcup\limits_{m=p-1}^{2^{n}-1}(U_{n}^{m}(l)\cap
f^{-1}\mathcal{O}_{n})\big{)}$ there are a neighborhood
$\mathcal{O}_{n+1}^{p-1}\subset\mathcal{O}_{n}$ of $y$, and open in
$f^{-1}\mathcal{O}_{n+1}^{p-1}$ subsets
$\widetilde{V}^{p-1}_{l}\subset\bigcup\limits_{m=p-1}^{2^{n}-1}U_{n}^{m}(l)$,
$l\in\mathbb{N}$, such that

$\displaystyle\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}^{p-1}}\big{(}\bigcup\limits_{m=p}^{2^{n}-1}(U_{n}^{m}(l)\cap
f^{-1}\mathcal{O}^{p-1}_{n+1}\big{)}\big{)}\subset\widetilde{V}^{p-1}_{l}\subset\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}^{p-1}}\widetilde{V}^{p-1}_{l}\subset$
(13)
$\displaystyle\subset\bigcup\limits_{i=l}^{\infty}\big{(}\bigcup\limits_{m=p-1}^{2^{n}-1}(U_{n}^{m}(l)\cap
f^{-1}\mathcal{O}_{n+1}^{p-1})\big{)},\ l\in\mathbb{N}.$

Thirdly, for the $F_{\sigma}$–subset
$\bigcup\limits_{l=1}^{\infty}\Big{(}T_{l}\cap f^{-1}\mathcal{O}_{n}\Big{)}$
of $f^{-1}\mathcal{O}_{n}$ and its neighborhood
$\bigcup\limits_{l=1}^{\infty}U_{n}^{2^{n}-1}(l)$ there are a neighborhood
$\mathcal{O}_{n+1}^{2^{n}-1}\subset\mathcal{O}_{n}$ of $y$ and open in
$f^{-1}\mathcal{O}_{n+1}^{2^{n}-1}$ subsets $\widetilde{V}^{2^{n}-1}(l)$,
$l\in\mathbb{N}$, such that

$\displaystyle T_{l}\cap f^{-1}\mathcal{O}_{n+1}^{2^{n}-1}\ \
\subset\widetilde{V}_{l}^{2^{n}-1}$ $\displaystyle\subset\ \
\mathrm{cl}_{\mathcal{O}_{n+1}^{2^{n}-1}}\widetilde{V}_{l}^{2^{n}-1}\subset$
(14)
$\displaystyle\subset\bigcup\limits_{l=1}^{\infty}\Big{(}U_{n}^{2^{n}-1}(l)\cap
f^{-1}\mathcal{O}_{n+1}^{2^{n}-1}\Big{)},\ l\in\mathbb{N}.$

Put $\mathcal{O}_{n+1}=\bigcap\limits_{p=0}^{2^{n}-1}\mathcal{O}_{n}^{p}$ ,
$V_{l}^{p}=\widetilde{V}_{l}^{p}\cap f^{-1}\mathcal{O}_{n+1}$,
$l\in\mathbb{N}$, $p\in\overline{0,2^{n}-1}$. Then for the families of sets
$\\{V_{l}^{p}\\}_{p=0}^{2^{n}-1}$, $l\in\mathbb{N}$, the following hold. For
any $l\in\mathbb{N}$

  * (i)

directly from the construction it follows that $V_{l}^{p}\supset V_{l}^{p+1}$,
$p\in\overline{0,2^{n}-2}$;

  * (ii)

from the inclusions (12) and the fulfillment of condition (b) for regular
$2^{n}$-partitions $\\{U^{k}_{n}(l)\\}_{k=0}^{2^{n}-1}$ of the subspace
$f^{-1}\mathcal{O}_{n}$ it follows that

$\displaystyle
F\cap\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}}V_{l}^{0}=\varnothing,$
$\displaystyle\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}}\Big{(}\bigcup\limits_{k=1}^{2^{n}-1}(U_{n}^{k}(l)\cap
f^{-1}\mathcal{O}_{n+1})\Big{)}\subset V_{l}^{0};$

  * (iii)

from the inclusions (13) it follows that for $p\in\overline{0,2^{n}-3}$

$\displaystyle(\bigcup\limits_{l=1}^{\infty}\bigcup\limits_{k=0}^{p}(U_{n}^{k}(l)\cap
f^{-1}\mathcal{O}_{n+1})\big{)}\cap\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}}V_{l}^{p+1}=\varnothing,$
$\displaystyle\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}}(\bigcup\limits_{k=p+2}^{2^{n}-1}(U_{n}^{k}(l)\cap
f^{-1}\mathcal{O}_{n+1})\big{)}\subset V_{l}^{p+1};$

  * (iv)

from the inclusions (14) and the fulfillment of condition (c) for a regular
$2^{n}$-partitions $\\{U^{k}_{n}(l)\\}_{k=0}^{2^{n}-1}$ of the subset
$f^{-1}\mathcal{O}_{n}$ it follows that

$\displaystyle(\bigcup\limits_{l=1}^{\infty}\bigcup\limits_{k=0}^{2^{n}-2}(U_{n}^{k}(l)\cap
f^{-1}\mathcal{O}_{n+1})\big{)}\cap\mathrm{cl}_{f^{-1}\mathcal{O}_{n+1}}V_{l}^{2^{n}-1}=\varnothing,$
$\displaystyle T_{l}\cap f^{-1}\mathcal{O}_{n+1}\subset V_{l}^{2^{n}-1}.$

Put $U_{n+1}^{2k}(l)=(U_{n}^{k}(l)\setminus V_{l}^{k})\cap
f^{-1}\mathcal{O}_{n+1}$, $U_{n+1}^{2k+1}(l)=(U_{n}^{k}(l)\cap V_{l}^{k})\cap
f^{-1}\mathcal{O}_{n+1}$, $k\in\overline{0,2^{n}-1}$, $l\in\mathbb{N}$. Then
from (ii) – (iv) it follows that $\\{U^{k}_{n+1}(l)|\
k\in\overline{0,2^{n+1}-1}\\}$ is a regular $2^{n+1}$–partition of the
subspace $f^{-1}\mathcal{O}_{n+1}$, $l\in\mathbb{N}$, for which condition of
the Definition 2.3 is fulfilled (i.e. $U_{n+1}^{2k}(l)\cup
U_{n+1}^{2k+1}(l)=U_{n}^{k}(l)\cap f^{-1}\mathcal{O}_{n+1},\
k\in\overline{0,2^{n}-1}$).

The constructed sequence of neighborhoods $\mathcal{O}_{n}$ and the families
of regular $2^{n}$-partitions of the subspace $f^{-1}\mathcal{O}_{n}$,
$n\in\omega_{0}$, are the consistent families
$\\{\mathcal{O}_{n},\\{U_{n}^{k}(l)|\ k\in\overline{0,2^{n}-1}\\}|\
n\in\mathbb{\omega}_{0}\\}$ of binary partitions of the mapping $f:X\to Y$ at
$y$ for each $l\in\mathbb{N}$. The fulfillment of properties (b) and (c)
follows from (ii) and (iv). ∎

######  Theorem 4.3.

For a mapping $f:X\to Y$ the following conditions are equivalent.

  * (A)

The mapping $f$ is $\sigma$–normal.

  * (B)

For any $\mathcal{O}\in\tau_{Y}$, any disjoint closed submapping
$f_{\mathcal{O}}|_{F}:F\to\mathcal{O}$ and $F_{\sigma}$-submapping
$f_{\mathcal{O}}|_{T}:T\to Y$ of the mapping
$f_{\mathcal{O}}:f^{-1}\mathcal{O}\to\mathcal{O}$ and any point
$y\in\mathcal{O}$ there exist consistent families of binary partitions
$\\{\mathcal{O}_{n},\\{U_{n}^{k}(l)|\ k\in\overline{0,2^{n}-1}\\}|\
n\in\omega_{0}\\}$, $l\in\mathbb{N}$, of the mapping $f:X\to Y$ at $y$ such
that for any $n\in\mathbb{N}$

    * (a)

$T\cap f^{-1}\mathcal{O}_{1}=\bigcup\limits_{l=1}^{\infty}T_{l}$ , where
$T_{l}$ is closed in $f^{-1}\mathcal{O}_{1}$, $l\in\mathbb{N}$;

    * (b)

$F\cap f^{-1}\mathcal{O}_{n}\subset U_{n}^{0}(l)$, $T_{l}\cap
f^{-1}\mathcal{O}_{n}\subset U_{n}^{2^{n}-1}(l)$, $l\in\mathbb{N}$;

    * (c)

$F\cap\mathrm{cl}_{f^{-1}\mathcal{O}_{n}}(\bigcup\limits_{k=1}^{2^{n}-1}U_{n}^{k}(l))=\varnothing,\
\
\mathrm{cl}_{f^{-1}\mathcal{O}_{n}}(\bigcup\limits_{k=0}^{2^{n}-2}U_{n}^{k}(l))\cap
T_{l}=\varnothing$, $l\in\mathbb{N}$.

  * (C)

For any $\mathcal{O}\in\tau_{Y}$ and any disjoint closed submapping
$f_{\mathcal{O}}|_{F}:F\to Y$ and $F_{\sigma}$–submapping
$f_{\mathcal{O}}|_{T}:T\to Y$ of the mapping
$f_{\mathcal{O}}:f^{-1}\mathcal{O}\to\mathcal{O}$ and any point
$y\in\mathcal{O}$ there exist a neighborhood $\mathcal{O}y$ of $y$ and a
$f$-equicontinuous at $y$ family of functions $\varphi_{l}:X\to[0,1]$,
$l\in\mathbb{N}$, such that

    * (a)

$T\cap f^{-1}\mathcal{O}y=\bigcup\limits_{n=1}^{\infty}T_{l}$, $T_{l}$ are
closed in $f^{-1}\mathcal{O}y$, $l\in\mathbb{N}$;

    * (b)

$\mathrm{osc}_{\varphi_{l}}(f^{-1}\mathcal{O}y)<\frac{1}{2}$,
$l\in\mathbb{N}$;

    * (c)

$F\cap f^{-1}\mathcal{O}y\subset\varphi_{l}^{-1}(0)\cap f^{-1}\mathcal{O}y$
and $T_{l}\cap f^{-1}\mathcal{O}y\subset\varphi_{l}^{-1}(1)\cap
f^{-1}\mathcal{O}y$, $l\in\mathbb{N}$;

    * (d)

$T_{l}\cap
f^{-1}\mathcal{O}y\subset\mathrm{int}_{f^{-1}\mathcal{O}y}(\varphi_{l}^{-1}(\frac{1}{2},1]\cap
f^{-1}\mathcal{O}y)$,
$\mathrm{cl}_{f^{-1}\mathcal{O}y}(\varphi_{l}^{-1}(\frac{1}{2},1]\cap
f^{-1}\mathcal{O}y)\cap F=\varnothing$, $l\in\mathbb{N}$.

###### Proof.

(A) $\Rightarrow$ (B) by Lemma 4.2.

(B) $\Rightarrow$ (C). Condition (B) implies the existence of the consistent
families of binary partitions $\\{\mathcal{O}_{n},\\{U^{k}_{n}(l)|\
k\in\overline{0,2^{n}-1}\\}|\ n\in\omega_{0}\\}$, $l\in\mathbb{N}$, of the
mapping $f:X\to Y$ at $y$ for which satisfy conditions (a) — (c) of (B).

By Proposition 2.3 for each $l\in\mathbb{N}$ by the consistent family of
binary partitions $\\{\mathcal{O}_{n},\\{U^{k}_{n}(l)|\
k\in\overline{0,2^{n}-1}\\}|\ n\in\omega_{0}\\}$, $l\in\mathbb{N}$, of the
mapping $f:X\to Y$ at $y$, a $f$-continuous at $y$ functions
$\varphi_{n}^{l}:X\to[0,1]$

$\varphi_{n}^{l}(x)=\left\\{\begin{array}[]{ll}0,&x\in X\setminus
f^{-1}\mathcal{O}_{n},\\\ \frac{k}{2^{n}-1},&x\in U^{k}_{n}(l),\
k\in\overline{0,2^{n}-1}\end{array}\right.$

are constructed. And they yield the construction of the family of
$f$-continuous at $y$ functions $\varphi_{l}:X\to[0,1]$

$\varphi_{l}(x)=\left\\{\begin{array}[]{ll}\varphi_{n}^{l}(x),&x\in
f^{-1}\mathcal{O}_{n}\setminus f^{-1}\mathcal{O}_{n+1},\
n\in\mathbb{\omega}_{0},\\\
\lim\limits_{n\to\infty}\varphi_{n}^{l}(x),&x\in\bigcap\limits_{n=0}^{\infty}f^{-1}\mathcal{O}_{n},\\\
\end{array}\right.$

such that ${\rm
osc}_{\varphi_{l}}(f^{-1}\mathcal{O}_{n})\leqslant\frac{1}{2^{n}-1}$ for all
$l\in\mathbb{N},\ n\in\omega_{0}$. Therefore, the family of functions
$\\{\varphi_{l}\\}$ is $f$-equicontinuous at $y$.

Moreover, firstly, for all $l\in\mathbb{N}$

$\varphi_{l}(x)=0$ for $x\in F$ (since $F\subset U_{n}^{0}(l)$,
$n\in\mathbb{N}$),

$\varphi_{l}(x)=1$ for $x\in T_{l}$ (since $T_{l}\subset U_{n}^{2^{n}-1}(l)$,
$n\in\mathbb{N}$).

Secondly, from the $f$-continuity of $\varphi_{l}$ the existence of a
neighborhood $\mathcal{O}y=\mathcal{O}_{2}$ of $y$ such that
$\mathrm{osc}_{\varphi_{l}}(f^{-1}\mathcal{O}y)<\frac{1}{2}$,
$l\in\mathbb{N}$, follows. Then by Lemma 1.1

$F\cap f^{-1}\mathcal{O}y\subset\varphi_{l}^{-1}(0)\cap
f^{-1}\mathcal{O}y\subset
f^{-1}\mathcal{O}y\setminus\mathrm{cl}_{f^{-1}\mathcal{O}y}(\varphi_{l}^{-1}[\tfrac{1}{2},1]\cap
f^{-1}\mathcal{O}y),$

and therefore, $T_{l}\cap
f^{-1}\mathcal{O}y\subset\mathrm{int}_{f^{-1}\mathcal{O}y}(\varphi_{l}^{-1}(\frac{1}{2},1]\cap
f^{-1}\mathcal{O}y)$, $l\in\mathbb{N}$. So $T_{l}\cap
f^{-1}\mathcal{O}y\subset U^{1}_{1}(l)\cap
f^{-1}\mathcal{O}y\subset\mathrm{int}_{f^{-1}\mathcal{O}y}(\varphi_{l}^{-1}[\frac{1}{2},1]\cap
f^{-1}\mathcal{O}y),\l\in\mathbb{N}$.

(C) $\Rightarrow$ (A). Consider an arbitrary disjoint closed subset $F$ and
$F_{\sigma}$–subset $T=\bigcup\limits_{l=1}^{\infty}T_{l}$ of the set
$f^{-1}\mathcal{O}$ for an arbitrary mapping
$f_{\mathcal{O}}:f^{-1}\mathcal{O}\to\mathcal{O}$. Let $y\in\mathcal{O}$ be
arbitrary. Then, it follows from item (d) of condition (C) that the
neighborhood $\mathcal{O}y$ of the point $y$ and the neighborhoods
$\mathrm{int}_{f^{-1}\mathcal{O}y}(\varphi_{l}^{-1}(\frac{1}{2},1]\cap
f^{-1}\mathcal{O}y)$ of the sets $T_{l}\cap f^{-1}\mathcal{O}y$,
$l\in\mathbb{N}$, are required. ∎

##  5 Perfect normality of a mapping.

######  Definition 5.1.

A mapping $f:X\to Y$ is perfectly normal if for any open subset $O\subset X$
and for any point $y\in Y$ there exist a neighborhood $\mathcal{O}y$ of $y$
and a $f$-equicontinuous at $y\in Y$ countable family of functions
$\\{\varphi_{l}:X\to[0,1]\ |\ l\in\mathbb{N}\\}$ such that

  * (1)

$O\cap
f^{-1}\mathcal{O}y=\bigcup\limits_{l\in\mathbb{N}}({\varphi_{l}}^{-1}(1)\cap
f^{-1}\mathcal{O}y),$

  * (2)

$f^{-1}\mathcal{O}y\setminus O\subset{\varphi_{l}}^{-1}(0)\cap
f^{-1}\mathcal{O}y,\ l\in\mathbb{N}.$

######  Remark 5.1.

In the Definition 5.1 one has.

  * (a)

One can assume that $O\cap
f^{-1}\mathcal{O}y=\bigcup\limits_{l\in\mathbb{N}}{\rm
cl}_{f^{-1}\mathcal{O}y}(\varphi_{l}^{-1}(1)\cap f^{-1}\mathcal{O}y)$.

  * (b)

The condition (2) is equivalent to the following one
$f^{-1}\mathcal{O}y\setminus
O=\bigcap\limits_{l\in\mathbb{N}}{\varphi_{l}}^{-1}(0)\cap
f^{-1}\mathcal{O}y$.

  * (c)

Any open submapping is a $F_{\sigma}$-submapping.

######  Proposition 5.1.

A submapping $f|_{X_{0}}:X_{0}\to Y$ of a perfectly normal mapping $f:X\to Y$
is perfectly normal.

###### Proof.

Consider an arbitrary submapping $f|_{X_{0}}:X_{0}\to Y$ of the mapping
$f:X\to Y$, a point $y\in Y$ and an open in $X_{0}$ subset $O$. There is an
open in $X$ subset $\widetilde{O}$ such that $O=\widetilde{O}\cap X_{0}$. From
the perfect normality of the mapping $f$ it follows that there is a
neighborhood $\mathcal{O}y$ of $y$ and a countable family of
$f$-equicontinuous at $y\in Y$ functions $\\{\varphi_{l}:X\to[0,1]\ |\
l\in\mathbb{N}\\}$, such that

$\widetilde{O}\cap
f^{-1}\mathcal{O}y=\bigcup\limits_{l\in\mathbb{N}}\varphi_{l}^{-1}(1)\cap
f^{-1}\mathcal{O}y,\
f^{-1}\mathcal{O}y\setminus\widetilde{O}\subset\varphi_{l}^{-1}(0)\cap
f^{-1}\mathcal{O}y,\ l\in\mathbb{N}.$

It is obvious that from the $f$-equicontinuity at $y\in Y$ of a countable
family of functions $\\{\varphi_{l}:X\to[0,1]\ |\ l\in\mathbb{N}\\}$ it
follows $f|_{X_{0}}$–equicontinuity at $y\in Y$ of a countable family of
functions $\\{\varphi_{l}|_{X_{0}}:X_{0}\to[0,1]\ |\ l\in\mathbb{N}\\}$.

Since $X_{0}\subset X$ and $\mathcal{O}y\subset Y$, then $\widetilde{O}\cap
X_{0}\cap f^{-1}\mathcal{O}y=O\cap(f|_{X_{0}})^{-1}\mathcal{O}y$ and
$X_{0}\cap\varphi_{l}^{-1}M=(\varphi_{l}|_{X_{0}})^{-1}M$, for
$M\subset[0,1]$, $l\in\mathbb{N}$. Hence

$\displaystyle O\cap(f|_{X_{0}})^{-1}\mathcal{O}y$
$\displaystyle=\bigcup\limits_{l\in\mathbb{N}}(\varphi_{l}|_{X_{0}})^{-1}(1)\cap
f^{-1}\mathcal{O}y,$ $\displaystyle(f|_{X_{0}})^{-1}\mathcal{O}y\setminus O$
$\displaystyle\subset(\varphi_{l}|_{X_{0}})^{-1}(0)\cap f^{-1}\mathcal{O}y,\
l\in\mathbb{N}.$

∎

######  Proposition 5.2.

A perfectly normal mapping $f:X\to Y$ is prenormal.

###### Proof.

Consider arbitrary disjoint subsets $F$ and $T$ closed in $X$ and a point
$y\in Y$.

From the perfect normality of the mapping $f$, for an open in $X$ subset
$U=X\setminus F$ there exist a neighborhood $\mathcal{O}^{\prime}y$ of $y$ and
a $f$-equicontinuous at $y$ family of functions
$\\{{\varphi}_{l}:X\to[0,1]\\}_{l\in\mathbb{N}}$ such that

$U\cap
f^{-1}\mathcal{O}^{\prime}y=\bigcup\limits_{l\in\mathbb{N}}{\varphi}_{l}^{-1}(1)\cap
f^{-1}\mathcal{O}y,\ F=f^{-1}\mathcal{O}^{\prime}y\setminus
U\subset{\varphi}_{l}^{-1}(0)\cap f^{-1}\mathcal{O}y\ l\in\mathbb{N}.$

Similarly, for an open in $X$ subset $V=X\setminus T$ there are a neighborhood
$\mathcal{O}^{\prime\prime}y$ of $y$ and a $f$-equicontinuous at $y$ family of
functions $\\{{\psi}_{l}:X\to[0,1]\\}_{l\in\mathbb{N}}$ such that

$V\cap
f^{-1}\mathcal{O}^{\prime\prime}y=\bigcup\limits_{l\in\mathbb{N}}{\psi}_{l}^{-1}(1)\cap
f^{-1}\mathcal{O}y,\ T=f^{-1}\mathcal{O}^{\prime\prime}y\setminus
V\subset{\psi}_{l}^{-1}(0)\cap f^{-1}\mathcal{O}y,\ l\in\mathbb{N}.$

Since the families of functions $\\{\varphi_{l}\\}_{l\in\mathbb{N}}$ and
$\\{\psi_{l}\\}_{l\in\mathbb{N}}$ are $f$-equicontinuous at $y$, there is a
neighborhood
$\mathcal{O}y\subset\mathcal{O}^{\prime}y\cap\mathcal{O}^{\prime\prime}y$ of
$y$ such that ${\rm osc}_{\varphi_{l}}(f^{-1}\mathcal{O}y)<\frac{1}{2}$ and
${\rm osc}_{\psi_{l}}(f^{-1}\mathcal{O}y)<\frac{1}{2}$, $l\in\mathbb{N}$.
Besides,

$\displaystyle F\cap
f^{-1}\mathcal{O}y\subset\bigcup\limits_{l\in\mathbb{N}}{\rm
int}_{f^{-1}\mathcal{O}y}(\psi_{l}^{-1}(\tfrac{1}{2},1]\cap
f^{-1}\mathcal{O}y)\subset\bigcup\limits_{l\in\mathbb{N}}{\rm
cl}_{f^{-1}\mathcal{O}y}(\psi_{l}^{-1}(\tfrac{1}{2},1]\cap
f^{-1}\mathcal{O}y),$ (15) $\displaystyle T\cap
f^{-1}\mathcal{O}y\subset\bigcup\limits_{l\in\mathbb{N}}{\rm
int}_{f^{-1}\mathcal{O}y}(\varphi_{l}^{-1}(\tfrac{1}{2},1]\cap
f^{-1}\mathcal{O}y)\subset\bigcup\limits_{l\in\mathbb{N}}{\rm
cl}_{f^{-1}\mathcal{O}y}(\varphi_{l}^{-1}(\tfrac{1}{2},1]\cap
f^{-1}\mathcal{O}y).$

From Lemma 1.1 and inclusions (15) we have

$\displaystyle F\cap f^{-1}\mathcal{O}y\cap\bigcup\limits_{l\in\mathbb{N}}{\rm
cl}_{f^{-1}\mathcal{O}y}(\varphi_{l}^{-1}(\tfrac{1}{2},1]\cap
f^{-1}\mathcal{O}y)$ $\displaystyle=\varnothing,$ $\displaystyle T\cap
f^{-1}\mathcal{O}y\cap\bigcup\limits_{l\in\mathbb{N}}{\rm
cl}_{f^{-1}\mathcal{O}y}(\psi_{l}^{-1}(\tfrac{1}{2},1]\cap
f^{-1}\mathcal{O}y)$ $\displaystyle=\varnothing.$

The disjoint closed in $f^{-1}\mathcal{O}y$ subsets $F\cap f^{-1}\mathcal{O}y$
and $T\cap f^{-1}\mathcal{O}y$, countable families of neighborhoods $\\{{\rm
int}_{f^{-1}\mathcal{O}y}(\varphi_{l}^{-1}(\tfrac{1}{2},1]\cap
f^{-1}\mathcal{O}y)\\}_{l\in\mathbb{N}}$ and $\\{{\rm
int}_{f^{-1}\mathcal{O}y}(\psi_{l}^{-1}(\tfrac{1}{2},1]\cap
f^{-1}\mathcal{O}y)\\}_{l\in\mathbb{N}}$ satisfy the normalizing Lemma [7,
Chapter 1, $\S$ 5, Lemma 2]. Thus, the subsets $F$ and $T$ are disjoint in
$f^{-1}\mathcal{O}y$. This implies that the mapping $f$ is prenormal. ∎

Since the perfect normality of a mapping is a hereditary property, and any
perfectly normal mapping is prenormal, then by [5, Proposition 7] we have.

######  Theorem 5.3.

A perfectly normal mapping $f:X\to Y$ is hereditarily normal.

The following definition is a variant of extending the concept of a
functionally open set to the case of mappings. It differs from the definition
at [1, p. 76] and allows to define the necessary condition of perfect
normality, similar to Vedenisov’s condition of perfect normality for spaces.

######  Definition 5.2.

A submapping $f|_{U}:U\to Y$ of a mapping $f:X\to Y$ is $f$–functionally open
if for any point $y$ there are a neighborhood $\mathcal{O}y$ of $y$ and a
$f$-continuous function $\varphi:X\to[0,1]$ such that $U\cap
f^{-1}\mathcal{O}y=\varphi^{-1}(0,1]\cap f^{-1}\mathcal{O}y$.

A submapping $f|_{F}:F\to Y$ of a mapping $f:X\to Y$ is $f$–functionally
closed if for any point $y$ there are a neighborhood $\mathcal{O}y$ of $y$ and
a $f$-continuous function $\varphi:X\to[0,1]$ such that $F\cap
f^{-1}\mathcal{O}y=\varphi^{-1}(0)\cap f^{-1}\mathcal{O}y$.

######  Remark 5.2.

A submapping $f|_{U}:U\to Y$ of a mapping $f:X\to Y$ is $f$-functionally open
iff the submapping $f|_{X\setminus U}:X\setminus U\to Y$ is $f$-functionally
closed.

######  Proposition 5.4.

Any open submapping $f|_{O}:O\to Y$ (closed submapping $f|_{F}:F\to Y$) of a
perfectly normal mapping $f:X\to Y$ is $f$-functionally open (respectively,
$f$-functionally closed).

###### Proof.

Consider an arbitrary point $y\in Y$ and an open in $X$ subset $O$. From the
perfect normality of the mapping $f$ it follows that there are a
$f$-equicontinuous countable family of functions $\\{\varphi_{l}:X\to[0,1]\ |\
l\in\mathbb{N}\\}$ at $y\in Y$ and a neighborhood $\mathcal{O}y$ of $y$ such
that

$O\cap
f^{-1}\mathcal{O}y=\bigcup\limits_{l\in\mathbb{N}}\varphi_{l}^{-1}(1)\cap
f^{-1}\mathcal{O}y,\ f^{-1}\mathcal{O}y\setminus
O\subset\varphi_{l}^{-1}(0)\cap f^{-1}\mathcal{O}y,\ l\in\mathbb{N}.$

The series
$\sum\limits_{l=1}^{\infty}\big{|}\big{|}\frac{\varphi_{l}}{2^{l}}\big{|}\big{|}_{f^{-1}\mathcal{O}y}$
converges. So, by Lemma 2.4 the function $\varphi:X\to[0,1]$

$\varphi(x)=\sum_{l=1}^{\infty}\frac{\varphi_{l}(x)}{2^{l}}$

is $f$-continuous at $y$. If $x\in O\cap f^{-1}\mathcal{O}$, then there exists
$l\in\mathbb{N}$ such that $x\in\varphi^{-1}(1)$. Then
$\varphi(x)\geqslant\frac{\varphi_{l}(x)}{2^{l}}=\frac{1}{2^{l}}>0$.
Therefore, $O\cap f^{-1}\mathcal{O}y=\varphi^{-1}(0,1]\cap f^{-1}\mathcal{O}y$
and the submapping is $f$–functionally open.

The proof of the $f$-functional closeness of a closed submapping follows from
the Remark 5.2. ∎

######  Definition 5.3.

A normal mapping $f:X\to Y$ is called a $co$-perfectly normal mapping if its
any open submapping $f|_{U}:U\to Y$, $U\in\tau_{X}$ is a
$F_{\sigma}$-submapping [3, Definition 8].

A $\sigma$-normal mapping $f:X\rightarrow Y$ is called a
$co$-$\sigma$-perfectly normal mapping if its any open submapping $f|_{U}:U\to
Y$, $U\in\tau_{X}$ is a $F_{\sigma}$-submapping [3, Definition 9].

Any $co$-$\sigma$-perfectly normal mapping is hereditarily normal [3, Theorem
14]. Any submapping of $co$-$\sigma$-perfectly normal mapping is
$co$-perfectly normal [3, Corollary 15].

######  Theorem 5.5 (a functional characterization of a $co$-$\sigma$-perfect
normality of a mapping).

A mapping $f:X\to Y$ is $co$-$\sigma$-perfectly normal iff for any
$\mathcal{O}\in\tau_{Y}$, for any open subset $U\subset f^{-1}\mathcal{O}$,
and for any $F_{\sigma}$-subset $F=\bigcup\limits_{l\in\mathbb{N}}F_{l}\subset
U$, where the sets $F_{l}$ are closed in $f^{-1}\mathcal{O}$,
$l\in\mathbb{N}$, for any point $y\in\mathcal{O}$ there are a neighborhood
$\mathcal{O}y$ of $y$ and a $f$-equicontinuous at $y$ family of functions
$\varphi_{l}:X\to[0,1]$ and $\psi_{l}:U\to[0,1]$, $l\in\mathbb{N}$, such that

  * (a)

$\mathrm{osc}_{\varphi_{l}}(f^{-1}\mathcal{O}y)<\frac{1}{2}$,
$\mathrm{osc}_{\psi_{l}}(f^{-1}\mathcal{O}y)<\frac{1}{2}$, $l\in\mathbb{N}$;

  * (b)

$f^{-1}\mathcal{O}y\setminus U\subset\varphi_{l}^{-1}(0)\cap
f^{-1}\mathcal{O}y$, $F_{l}\cap
f^{-1}\mathcal{O}y\subset\varphi_{l}^{-1}(1)\cap f^{-1}\mathcal{O}$,
$l\in\mathbb{N}$;

  * (c)

$f^{-1}\mathcal{O}y\setminus U\subset\psi_{l}^{-1}(0)\cap f^{-1}\mathcal{O}$,
$U\cap f^{-1}\mathcal{O}y=\bigcup\limits_{l\in\mathbb{N}}\psi^{-1}_{l}(1)\cap
f^{-1}\mathcal{O}$, $l\in\mathbb{N}$.

###### Proof.

Necessity. Without loss of generality, we will assume that $Y=\mathcal{O}$.
Let fix an arbitrary point $y\in Y$. From the $co$-$\sigma$-perfect normality
of the mapping $f$ it follows that its open submapping $f|_{U}:U\to Y$ has
type of $F_{\sigma}$. It means that there is a neighborhood
$\mathcal{O}^{\prime}y$ of $y$ such that $f^{-1}\mathcal{O}^{\prime}y\cap
U=\bigcup\limits_{l=1}^{\infty}T^{\prime}_{i}$, where $T^{\prime}_{l}$ is
closed in $f^{-1}\mathcal{O}^{\prime}y$ for all $l\in\mathbb{N}$.

For a countable family of closed subsets $T^{\prime}_{l}$,
$F^{\prime}_{l}=f^{-1}\mathcal{O}^{\prime}y\cap F_{l}$, $l\in\mathbb{N}$, we
have

$\bigcup\limits_{l=1}^{\infty}(T^{\prime}_{l}\cup
F^{\prime}_{l})=f^{-1}\mathcal{O}^{\prime}y\cap
U\Longrightarrow\bigcup\limits_{l=1}^{\infty}(T^{\prime}_{l}\cup
F^{\prime}_{l})\cap\big{(}f^{-1}\mathcal{O}^{\prime}y\setminus
U\big{)}=\varnothing.$

The set $f^{-1}\mathcal{O}^{\prime}y\setminus U$ is closed in
$f^{-1}\mathcal{O}^{\prime}y$. The mapping $f:X\to Y$ is $\sigma$-normal. By
Theorem 4.3 there exist a neighborhood
$\mathcal{O}y\subset\mathcal{O}^{\prime}y$ of $y$ and a $f$-equicontinuous at
$y$ family of functions $\varphi_{l}:X\to[0,1]$, $l\in\mathbb{N}$,
$\psi_{l}:X\to[0,1]$, $l\in\mathbb{N}$, such that

  * (a′)

$\mathrm{osc}_{\varphi_{l}}(f^{-1}\mathcal{O}y)<\frac{1}{2}$,
$\mathrm{osc}_{\psi_{l}}(f^{-1}\mathcal{O}y)<\frac{1}{2}$, $l\in\mathbb{N}$;

  * (b′)

$U\cap f^{-1}\mathcal{O}y=\bigcup\limits_{l=1}^{\infty}T_{l}$, where the sets
$T_{l}=T^{\prime}_{l}\cap f^{-1}\mathcal{O}y$ are closed in
$f^{-1}\mathcal{O}y$, and the sets $F_{l}\cap f^{-1}\mathcal{O}y\subset U\cap
f^{-1}\mathcal{O}y$, $l\in\mathbb{N}$, are closed in $f^{-1}\mathcal{O}y$;

  * (c′)

$f^{-1}\mathcal{O}y\setminus U\subset\varphi_{l}^{-1}(0)\cap
f^{-1}\mathcal{O}y$, $f^{-1}\mathcal{O}y\setminus U\subset\psi_{l}^{-1}(0)\cap
f^{-1}\mathcal{O}y$,

$F_{l}\cap f^{-1}\mathcal{O}y\subset\varphi_{l}^{-1}(1)\cap
f^{-1}\mathcal{O}y$, $T_{l}\cap f^{-1}\mathcal{O}y\subset\psi_{l}^{-1}(1)\cap
f^{-1}\mathcal{O}y$, $l\in\mathbb{N}$.

From the conditions (a′) — (c′) the conditions of the theorem follows.

Sufficiency. Let us show that any open submapping of the mapping $f:X\to Y$ is
of type $F_{\sigma}$. Without loss of generality, we will assume that
$Y=\mathcal{O}$. The set $U$ is open in $X$.

According to conditions of the theorem, for any point $y\in Y$ there exist its
neighborhood $\mathcal{O}y$ and a $f$-equicontinuous at $y$ family of
functions $\psi_{l}:f^{-1}\mathcal{O}y\to[0,1]$, $l\in\mathbb{N}$, such that:

$\displaystyle\mathrm{osc}_{\psi_{l}}(f^{-1}\mathcal{O}y)<\tfrac{1}{2},\
f^{-1}\mathcal{O}y\setminus U\subset\psi_{l}^{-1}(0)\cap f^{-1}\mathcal{O}y,\
l\in\mathbb{N},\text{ and }$ $\displaystyle U\cap
f^{-1}\mathcal{O}y=\bigcup\limits_{l\in\mathbb{N}}\psi^{-1}_{l}(1)\cap
f^{-1}\mathcal{O}y.$

By Lemma 1.1 (for $a=0$, $b=1$) we have $\\{x\in
f^{-1}\mathcal{O}y\;|\;\psi_{l}(x)=0\\}\cap\mathrm{cl}_{f^{-1}\mathcal{O}y}\\{x\in
f^{-1}\mathcal{O}y\;|\;\psi_{l}(x)=1\\}=\varnothing$. Let
$T_{l}=\mathrm{cl}_{f^{-1}\mathcal{O}y}\\{x\in
f^{-1}\mathcal{O}y\;|\;\psi_{l}(x)=1\\}$, $l\in\mathbb{N}$. Then, $U\cap
f^{-1}\mathcal{O}y=\bigcup\limits_{l\in\mathbb{N}}T_{l}$ and the mapping $f$
is of type $F_{\sigma}$.

Let’s show that the mapping $f:X\to Y$ is $\sigma$-normal. Without loss of
generality, we will assume that $Y=\mathcal{O}$. Let
$F=\bigcup\limits_{l\in\mathbb{N}}F_{l}$ be a $F_{\sigma}$-subset of $X$,
$F_{l}$ are closed in $X$, $T$ is closed in $X$ and $T\cap F=\varnothing$.
Then $F\subset U=X\setminus T$.

From the conditions of the theorem it follows that for any point $y\in Y$
there exist its neighborhood $\mathcal{O}y$ and a $f$-equicontinuous at $y$
family of functions $\varphi_{l}:X\to[0,1]$, $l\in\mathbb{N}$, such that

$\displaystyle\mathrm{osc}_{\varphi_{l}}(f^{-1}\mathcal{O}y)<\tfrac{1}{2},\
F_{l}\cap f^{-1}\mathcal{O}y\subset\varphi_{l}^{-1}(1)\cap
f^{-1}\mathcal{O}y,$ $\displaystyle f^{-1}\mathcal{O}y\setminus
U\subset\varphi_{l}^{-1}(0)\cap f^{-1}\mathcal{O}y,\ l\in\mathbb{N}.$

By Theorem 4.3 the mapping $f:X\to Y$ is $\sigma$-normal. ∎

######  Corollary 5.6.

For a $\sigma$-normal mapping $f:X\to Y$ the following conditions are
equivalent.

  * (A)

The mapping $f:X\to Y$ is $co$-$\sigma$-perfectly normal.

  * (B)

For any open submapping $f|_{U}:U\to Y$ of $f$ and any point $y\in Y$ there
are a neighborhood $\mathcal{O}y$ of $y$ and a $f$-equicontinuous at $y$
family of functions $\psi_{l}:X\to[0,1]$, $l\in\mathbb{N}$, such that

    * (1)

$\mathrm{osc}_{\psi_{l}}(f^{-1}\mathcal{O}y)<\frac{1}{2}$, $l\in\mathbb{N}$;

    * (2)

$f^{-1}\mathcal{O}y\setminus U\subset\psi_{l}^{-1}(0)\cap f^{-1}\mathcal{O}y$,
$l\in\mathbb{N}$, $U\cap
f^{-1}\mathcal{O}y=\bigcup\limits_{l\in\mathbb{N}}\psi^{-1}_{l}(1)\cap
f^{-1}\mathcal{O}y$.

From Theorem 5.5 and point (c) of Remark 5.1 we have.

######  Proposition 5.7.

Any $co$-$\sigma$-perfectly normal mapping is perfectly normal. Any perfectly
normal mapping is $co$-perfectly normal.

######  Remark 5.3.

In the case of a constant mapping, $co$-perfect normality, perfect normality
and $co$-$\sigma$-perfect normality of the mapping coincide.

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