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\title{Topologizing different classes of real functions} 
\author{}
\date{}

\markright{Topologizing different classes of real functions 
%\ \ \ \ \today
}


\begin{document}
\maketitle

{\small\noindent Krzysztof Ciesielski, Department of Mathematics, West
Virginia University, Morgantown, WV 26506-6310
  \footnote{
  Key Words: topologies on $\smallreals^n$, 
  linear, analytic and harmonic functions, 
  derivatives \\
  AMS Subject Classification. Primary:  26A15; Secondary 54H10, 03E35.
       }
}
 
\begin{abstract}
The purpose of this paper is to examine which classes \F\ of
functions from $\reals^n$ into $\reals^m$ 
can be topologized in a sense that
there exist topologies $\tau_1$ and $\tau_2$ on $\smallreals^n$ and
$\smallreals^m$, respectively, 
such that
\F\ is equal to the class $\C(\tau_1,\tau_2)$ 
of all continuous functions 
$f\colon(\smallreals^n,\tau_1)\to(\smallreals^m,\tau_2)$. 
We will show that the Generalized Continuum Hypothesis GCH
implies the positive answer for this question for a large
number of classes of functions \F\ for which the sets 
$\{x\colon f(x)=g(x)\}$ are small in some sense for all
$f,g\in\F$, $f\neq g$. The topologies will be Hausdorff and connected.
It will be also shown that in some model of set theory ZFC with GCH
these topologies could be completely regular and Baire.
One of the corollaries of this theorem is that
GCH implies the existence of a connected Hausdorff 
topology \T\ on \smallreals\ 
such that the class \Lin\ of all linear functions $g(x)=ax+b$
coincides with $\C(\T,\T)$.
This gives an affirmative answer to a question of Sam Nadler. 
The above corollary remains true for 
the class $\cal P$ of all polynomials,
the class \A\ of all analytic functions
and the class of all harmonic functions.

We will also prove that several other classes of real functions
cannot be topologized. This includes 
the classes of \cinfinity\ functions, differentiable functions,
Darboux functions and derivatives.

\end{abstract}
 
\section{$\!\!\!\!\!\!\!${\bf.} Introduction.}

There are a number of known classes of real functions that can be represented 
as 
families of continuous functions 
\[
\C(\tau_1,\tau_2)=
\{f\colon(\reals,\tau_1)\to(\reals,\tau_2)\colon f\mbox{ is continuous}\},
\] 
where $\tau_1$ and $\tau_2$ are the topologies on \reals. Evidently, the 
ordinary continuous functions are $\C(\ordinarytop,\ordinarytop)$, 
where \ordinarytop\ stands for the ordinary topology. The other 
obvious examples include the class 
$\C(\ordinarytop,\{(a,\infty)\colon a\in[-\infty,\infty]\})$ of 
lower semicontinuous functions, 
the class $\C(\ordinarytop,\{(-\infty,a)\colon a\in[-\infty,\infty]\})$ of 
upper semicontinuous functions and the class 
$\C(\tau_r,\ordinarytop)$ of right continuous functions, where topology
$\tau_r$ is generated by intervals $[a,b)$.
Probably the most interesting non-trivial example of topologized class
consists of the class \appr\ of approximately continuous functions.
This class was introduced by Denjoy in 1915
\cite{Denjoy:nombresDerives} and 
since then it was extensively studied,
including Zahorski's very deep work on the derivatives 
\cite{ZZ:PremDer} from 1950. However, it was not until 1952 when the 
density topology
\densitytop\ on \reals\ and the relation 
$\appr=\C(\densitytop,\ordinarytop)$
was discovered by Haupt and Pauc \cite{HauptPauc:topologie}.
Moreover, the paper of Haupt and Pauc was
completely unnoticed for years, and the real study of the density 
topology dates
from 1961 when Goffman and Waterman \cite{GW:AppContTrans}
rediscovered the density topology and
the relation $\appr=\C(\densitytop,\ordinarytop)$.

In the last two decades another approach was used: several classes of 
real functions were introduced as classes of functions continuous in some
topologies on \reals. This includes the classes:
$\C(\densitytop,\densitytop)$ of the density continuous functions, 
$\C(\itopology,\ordinarytop)=\C(\deepitopology,\ordinarytop)$ 
of the \I-approximately continuous functions,
$\C(\itopology,\itopology)$ of the \I-density continuous functions, and
$\C(\deepitopology,\deepitopology)$ of the deep \I-density continuous 
functions,
where the \I-density topology \itopology\ is considered to be a 
category analog
of the density topology, and deep \I-density topology \deepitopology\
is defined as the coarsest
topology for the \I-approximately continuous functions. (For more 
information on the subject, see \cite{CLO:book}. Compare also 
\cite{CL:Examples,GNN:densitytop,Oxtoby:MeasCat,Wilczynski:CatAn}.)

The purpose of this paper is to examine which of the 
other known classes of real functions can be topologized. 
This problem, with the emphasis on the classes of linear 
and differentiable
functions, was first articulated by Sam Nadler
in a student's problem session held at West Virginia University.
The author discussed this for the class of
linear functions with several people.
This version of the problem was also
restated by Lee Larson on 
Fifteen Summer Symposium in Real Analysis in Bratislava.

\medskip

The paper is organized as follows. Section \ref{sec:basic}
contains a discussion of relations between topologized class 
$\F=\C(\tau_1,\tau_2)$ and topologies $\tau_1$ and $\tau_2$. 
It also contains the proofs that several classes of real 
functions cannot be topologized.
Section \ref{SecMainTheorem} states the main theorem and
discuss its corollaries. The two cases of the theorem are proved separately.
The first part, which involves only the use of GCH and transfinite
induction, is presented in Section \ref{SecA}. The second part of 
the theorem, which proof involves forcing method, is left for 
Section \ref{SecB}, the last
section of the paper. The reader unfamiliar with the forcing technic
can simply skip this section.
Section \ref{SecDisc} contains a discussion of set theoretical assumptions
of the main theorem and points several possible generalizations. 

\medskip

The set theoretical and topological
terminology and notation used in this paper is 
standard and follows \cite{Kunen,Engelking:GenTop}.
In particular, ordinals are identified with their sets of predecessors
and cardinals with the initial ordinals. Symbol $\omega$ denotes
the first infinite ordinal as well as the first infinite cardinal.
${\cal P}(X)$ will denote the power set of $X$ and $|X|$ the 
cardinality of $X$. 
If $\kappa$ is a cardinal number than $\kappa^+$ denotes the 
cardinal successor of $\kappa$ and $2^\kappa=|{\cal P}(\kappa)|$. 
GCH will stand for the Generalized Continuum Hypothesis, i.e., the statement
that $2^\kappa=\kappa^+$ for every infinite cardinal $\kappa$.
The functions will be identified with 
their graphs. The class of all functions $f: X\to Y$
from a set $X$ to a set $Y$ is denoted by $Y^X$.
For $f: X\to Y$ the restriction of $f$ to a set 
$A\subset X$ will be denoted by $f|_A$.
For a set $X$ and a cardinal number $\kappa$ we define 
$[X]^{\leq\kappa}=\{Y\subset X: |Y|\leq\kappa\}$ and 
$[X]^{<\kappa}=\{Y\subset X: |Y|<\kappa\}$.
A family $\I\subset {\cal P}(X)$ is said to be an ideal
if $A\cup B\in\I$ provided $A,B\in\I$, and $B\in\I$ 
provided $B\subset A\in\I$.
An ideal \I\ is said to be a $\sigma$-ideal if $\bigcup\F\in\I$
for every $\F\in[\I]^{\leq\omega}$.

The letters \T\ and $\tau$, with possible subscripts, will 
always denote the
topologies. In particular, 
$\T_{\cal O}^n$, or simply
$\ordinarytop$, will denote
the ordinary topology on $\reals^n$. 
For topological spaces
$(X,\tau_1)$ and $(Y,\tau_2)$ the class of all continuous functions 
from $(X,\tau_1)$ to $(Y,\tau_2)$ is denoted by 
$\C(\tau_1,\tau_2)$. We will also write 
$C(\tau)$ in place of $C(\tau,\tau)$. 
Symbol $\const$ will denote the family of
all constant functions in currently considered class $Y^X$.
Symbol $id_X$ will stand for the identity
function on $X$ and dom$(f)$ for the domain of a function $f$.  
The closure and interior of a set $A$ in topology $\tau$ will be denoted
by cl$_\tau(A)$ and int$_\tau(A)$, respectively.

Symbol $\cal A$ will be reserved for the class of all real 
or complex analytic functions. ${\cal P}\subset{\cal A}$ and 
$\Lin\subset{\cal A}$ will stand for the class of all polynomials and 
the class of all linear functions $f(x)=ax+b$, respectively.
 
The definitions of these classes and other classes of real functions used
in this paper can be found in \cite{Bruckner:DiffReal} or in 
\cite{CLO:book,Wilczynski:CatAn}. 

\section{$\!\!\!\!\!\!\!${\bf.} Basic properties
of topologized classes and their applications.}\label{sec:basic}

In this section we will examine some of the properties of the topologies
$\tau_1$ and $\tau_2$ that can be deduced from the properties 
of $\C(\tau_1,\tau_2)$. This will give us a perspective 
on the difficulties we must face topologizing different classes
of real functions. Moreover, the results presented in this section 
will show the boundaries of the technic presented in the sections to follow.

The following theorem lists some basic properties of 
classes of functions that can be topologized. 
Recall also that a family $\T$ of subsets of \reals\ or \complexnumb\
is homothetically closed if $L^{-1}(U)\in\T$ for every
$L\in\Lin$ and $U\in\T$.

\theorem{th:properties}{ Let $\tau_1$ and $\tau_2$ be the 
topologies on sets $X$ and $Y$, respectively, and let
$\F=\C(\tau_1,\tau_2)\neq Y^X$. If
$\tau$ is a weak topology on $X$ generated by \F, i.e., 
generated by the family $\{f^{-1}(U)\colon U\in\tau_2, f\in\F\}$,
then
\begin{description}
\item[(i)]   $\const\subset\F$, $\tau\subset\tau_1$, 
             $\tau_1\neq{\cal P}(X)$,
             $\tau_2\neq\{\emptyset,Y\}$ and $\F=\C(\tau,\tau_2)$;

\item[(ii)]  if $X=Y$ and $id_X\in\F$ 
             then $\tau_2\subset\tau\subset\tau_1$;

\item[(iii)] if $Y=\reals$ and $\ordinarytop\subset\tau_2$ then \F\ is closed 
             under the maximum and minimum operations;

\item[(iv)]  if $\G\subset Y^Y$ is such that 
             $id_Y\in\G$ and $g\circ f\in\F$ 
             for all $f\in\F$ and $g\in\G$ then 
             $\F=\C(\tau,\tau^\prime)$,
             where $\tau^\prime$ is a topology generated by 
             $\{g^{-1}(U)\colon U\in\tau_2, g\in\G\}$; 

             in particular, if $\G=\Lin$ than we may assume that
             $\tau_2$ is a homothetically closed $T_1$ topology;

\item[(v)]   if $X=Y$, $id_X\in\F$ and \F\ is closed under the composition,
             then $\F=\C(\tau)$;

\item[(vi)]  if $X=Y\in\{\reals,\complexnumb\}$ and $\Lin\subset\F$
             then $\tau_1$ is a $T_1$ topology;

\item[(vii)] if $X=Y\in\{\reals,\complexnumb\}$, $\Lin\subset\F$
             and $\tau_2$ contains two nonempty disjoint sets, 
             then $\tau_1$ is Hausdorff;


\item[(viii)] if $X=Y=\reals$ and $\F\subset Darboux$ 
             then $\tau_1$ is connected;

\item[(ix)]  if $X=Y=\reals$, $\Lin\subset\F$ and $\tau_2$ contains a 
             nonempty set
             which has either upper or lower bound, then 
             $\ordinarytop\subset\tau_1$;
\end{description}
}

Proof. (i) and (ii) are obvious.

(iii) Let $f,g\in\F$. 
We have to prove that $\max\{f,g\}\in\F$.

First notice that the function 
$\vee\colon(\reals\times\reals,\tau_2\times\tau_2)\to(\reals,\tau_2)$
defined by $\vee(y,z)=\max\{y,z\}$ is continuous.
It follows from that fact for every $U\in\tau_2$ we have
\[
\vee^{-1}(U) =  (U\times U)
            \cup(\{(y,z)\colon y<z\}\cap(\reals\times U))
            \cup(\{(y,z)\colon y>z\}\cap(U\times\reals))
\]
and the set on the right hand side belongs to $\tau_2\times\tau_2$.
It is also easy to see that the following functions are continuous: 
$\triangle\colon(X,\tau_1)\to(X\times X,\tau_1\times\tau_1)$, 
$\triangle(x)=(x,x)$ and 
$f\times g\colon(X\times X,\tau_1\times\tau_1)\to
(\reals\times\reals,\tau_2\times\tau_2)$, 
$(f\times g)(x_1,x_2)=(f(x_1),g(x_2))$.
But $\max\{f,g\}=\vee\circ(f\times g)\circ\triangle$, so it is continuous as
a composition of continuous functions.

The argument for $\min\{f,g\}\in\F$ is essentially the same.

(iv) Evidently, by (i), 
$\C(\tau,\tau^\prime)\subset\C(\tau,\tau_2)=\C(\tau_1,\tau_2)=\F$,
since $\tau_2\subset\tau^\prime$. 
So, let $f\in\F$. The topology $\tau^\prime$ is generated by the 
sets of the form $g^{-1}(U)$ where $g\in\G$ and 
$U\in\tau_2$.
But $f^{-1}(g^{-1}(U))=
(g\circ f)^{-1}(U)\in\tau$, since $g\circ f\in\F$.
Hence, $\F\subset\C(\tau,\tau^\prime)$. 

Now, if $\G=\Lin$ then $\tau^\prime$ is clearly homothetically closed and
it must be a $T_1$ topology, since $\tau_2\neq\{\emptyset,Y\}$.
(Compare also (vi) below.)

(v)  By (iv) used with $\G=\F$
we have $\F=\C(\tau,\tau^\prime)$, where $\tau^\prime\subset\tau$.

(vi) Since $\tau_2\neq\{\emptyset,Y\}$, there are
$a,b\in Y$ and $U\in\tau_2$ such that  $a\in U$ and $b\not\in U$.
But the family
$\{f^{-1}(U)\colon U\in\tau_2, f\in\Lin\}\subset\tau\subset\tau_1$ 
is homothetically closed. So, $\tau_1$ is a $T_1$ topology.

(vii) is also implied by the fact that 
$\{f^{-1}(U)\colon U\in\tau_2, f\in\Lin\}\subset\tau_1$ 
is homothetically closed.

(viii) is true since the characteristic function of any clopen 
set is continuous.

(ix) is clear, since 
$\{f^{-1}(U)\colon U\in\tau_2, f\in\Lin\}\subset\tau\subset\tau_1$ 
is homothetically closed.

Theorem \ref{th:properties} is proved.

\bigskip

As an immediate corollary we obtain the following.

\corollary{cor:sameTop}{ Let $\F\subset X^X$.
If \F\ can be topologized then $\F=\C(\tau)$ for some topology $\tau$
on $X$ if and only if $id_X\in\F$ and 
\F\ if closed under the composition operation.}

Proof. If $\F=\C(\tau)$ then obviously \F\ is closed under composition
and $id_X\in\F$.
The other implication follows from Theorem \ref{th:properties}(v).

\bigskip

The main goal of the paper is to prove that a wide range of classes
of functions can be topologized. 
It is clear from Theorem \ref{th:properties}(i) that to fulfill this
project it is enough to construct only a topology $\tau_2$ on
the range of the class of functions. 
By Corollary \ref{cor:sameTop}
this is true even is we like to have the same topology on the domain
and the range. Moreover, 
by Theorem \ref{th:properties}(vi) and (viii), 
any topology on \reals\ topologizing class \Lin\ must be
$T_1$ and connected. Condition (vii) 
of Theorem \ref{th:properties} suggest also that it is wise
to construct this topology as Hausdorff. 
These are the properties that topologies constructed in the next sections will
have. Right now, let us list some of the immediate corollaries of
Theorem \ref{th:properties}.

\theorem{Cinfinity}{ Let \F\ be a family of real
functions closed under composition and such that 
$\cinfinity\subset\F$.
If \F\ can be topologized then \F\ is closed under 
the maximum and minimum operations.
}

Proof. Assume that \F\ can be topologized. Then, by 
Corollary \ref{cor:sameTop}, $\F=\C(\tau)$. Moreover, by 
Theorem \ref{th:properties}(vi), $\tau$ is 
$T_1$. Let $f(x)=e^{-x^{-2}}$ for
$x>0$ and $f(x)=0$ for $x\leq 0$. It is well known that 
$f\in\cinfinity$, so that
$f\in\F$. Hence, 
$(0,\infty)=f^{-1}(\reals\setminus\{0\})\in\tau$, since
$\reals\setminus\{0\}\in\tau$. But conditions (ix) and (iii) of
Theorem \ref{th:properties} imply now that 
$\ordinarytop\subset\tau$ and
that \F\ is closed under the maximum and minimum operations.

\bigskip

Theorem \ref{Cinfinity} implies the following corollary. The definitions
of most of the classes of the corollary can be found in 
\cite{Bruckner:DiffReal}
and \cite{Wilczynski:CatAn}.

\corollary{cor:Cinf}{ The classes: \cinfinity, 
${\cal D}^n$ of $n$-times differentiable 
functions and $\C^n$ of functions with continuous $n$-th 
derivative cannot be topologized.
The same is true, when in the above we replace differentiability with
symmetric differentiability, approximate differentiability,
symmetric approximate differentiability, 
\I-approximate differentiability or 
symmetric \I-approximate differentiability.}

Proof. All these classes contain \cinfinity\ as a subclass and 
are closed under the 
composition. Moreover, they are not closed under the maximum operation.
It is shown by 
the function $|x|=\max\{x,-x\}$ for all classes 
but those with prefix symmetric. 
To see that the symmetric classes are not closed under 
supremum, take $a_0>b_0>a_1>b_1>\ldots$ such that 
$\lim_n a_n=0$, $0$ is
a dispersion and \I-dispersion point of 
$\bigcup_{n\in\smallnats}[a_{n+1},b_n]$ 
and $0$ is
neither right density nor right \I-density point of 
$P_i=\bigcup_{n\in\smallnats}[b_{2n+i},a_{2n+i}]$ for $i<2$.
Define $f(0)=0$, $f(x)=(-1)^ix$ for $x\in P_i$, extend it to a 
\cinfinity\ function on
$(0,\infty)$ staying between the graphs of $x$ and $-x$ and define
$f(x)=f(-x)$ for $x<0$. It is clear that function $f$ 
is symmetrically
infinitely many times differentiable in any of the sense 
define in the theorem.
However, $h(x)=\max\{x,f(x)\}$ is in neither of these classes.

\bigskip

Before we state the next theorem let us define the following 
class of functions.
A function $f$ is in the class $\E_0$ provided there exists a sequence
\[
\infty\geq d_1>c_1>b_1>a_1>d_2>c_2>b_2>a_2>\ldots
\]
such that the right hand side density of 
the sets $P_f^1=\bigcup_{n\in\smallnats}[a_n,b_n]$
and $P_f^2=\bigcup_{n\in\smallnats}[c_n,d_n]$ with respect to $0$ 
is equal to $1/2$
and that $f$ is defined as follows: $f(x)=-1$ for $x\in P_f^1$, 
$f(x)=1$ for $x\in P_f^2$, $f$ is linear on each of the intervals 
$[b_n,c_n]$ and
$[d_{n+1},a_n]$, $f(0)=0$ and $f(x)=f(-x)$ for $x\in(-\infty,0)$.

\theorem{derivatives}{ If \F\ is a family of real functions such that
$\E_0\subset\F\neq\RtoR$ and $L\circ f\circ M\in\F$ for all 
$f\in\F$ and $L,M\in\Lin$, then
\F\ cannot be topologized.
}

Proof. By way of contradiction assume that $\F=\C(\tau_1,\tau_2)$. 
By Theorem \ref{th:properties}(iv) we may assume, without loss of 
the generality, 
that $\tau_2$ is a $T_1$ topology
and that $\tau_1$ is a weak topology generated by \F\ and $\tau_2$.
Choose $f,g\in\E_0$ such that 
\[
(0,\infty)=P_f^1\cup P_f^2\cup P_g^1\cup P_g^2
\subset f^{-1}(\{-1,1\})\cup g^{-1}(\{-1,1\}).
\]
Then, 
$\{0\}=f^{-1}(\reals\setminus\{-1,1\})
\cap g^{-1}(\reals\setminus\{-1,1\})\in\tau_1$.
But $\tau_1$ is homothetically closed, since for every 
$f\in\F$, $U\in\tau_2$ and $M\in\Lin$ we have 
$M^{-1}(f^{-1}(U))=(f\circ M)^{-1}(U)\in\tau_1$.
So, $\tau_1={\cal P}(\reals)$
and $\F=\RtoR$, contradicting our assumption.

\corollary{cor:der}{ The following classes cannot be topologized: 
the class of all
derivatives, the Zahorski's classes ${\cal M}_i$ for $i=0,1,2,3,4$, 
the class of all 
symmetrically 
(symmetrically approximately or symmetrically \I-approximately) 
continuous functions,
the class of all Darboux functions, the class of all measurable functions
and the class of all functions having the Baire property.}

Proof. It is easy to see that all the above classes are proper 
subclasses of \RtoR\
and are closed under interior and exterior compositions with 
linear functions.
The proof that all the
functions from the class $\E_0$ are derivatives and in class 
${\cal M}_4$ can be found
in \cite[pages 23 and 87]{Bruckner:DiffReal}. 
They are also evidently symmetrically
continuous. The rest follows from Theorem \ref{derivatives}.

\section{$\!\!\!\!\!\!\!${\bf.} Main theorem and its 
corollaries.}\label{SecMainTheorem}

In this section we state the main theorem and conclude from it some
corollaries. The proof of the theorem will be postponed until 
the next sections. Recall that family $\F\subset Y^X$ {\em separates points}
if for every distinct points $x_1,x_2\in X$ there is $f\in\F$ such 
that $f(x_1)\neq f(x_2)$.
The topological space is {\em Baire} if every first category set in
this topology has empty interior. 

Let us also stress here that in what follows regular and completely regular
topological spaces do not have to be Hausdorff. 

\theorem{main}{ Let $|X|=|Y|=\continuum$, $\R\in[Y^X]^{\leq 2^\omega}$
and let \I\
be a proper $\sigma$-ideal on $X$ containing all singletons.

{\rm{\bf (A)}} If GCH holds
then there is a Hausdorff, connected and
locally connected topology $\tau_2$ on $Y$
with the property that for every family $\F\subset\const\cup\R$ 
such that $\const\subset\F$ and
\begin{equation}\label{conThmain}
\{x\in X\colon f(x)=g(x)\}\in\I
\ \mbox{ for every distinct }\ f,g\in\F 
\end{equation}
we have
\[
\F=\C(\tau,\tau_2), 
\]
where $\tau$ is generated by the family
$\{f^{-1}(U)\colon U\in\tau_2, f\in\F\}$. 
Topology $\tau$ is connected and locally connected. It is also Hausdorff, 
provided \F\ separates points.

{\rm{\bf (B)}}  Moreover, it is consistent with the set theory ZFC 
plus GCH that the topologies $\tau$ and $\tau_2$ are completely regular
and Baire. 
}

To see the real meaning of Theorem \ref{main} let us state several of
its corollaries. 
In the first corollary we use Theorem \ref{main} with the 
$\sigma$-ideal $\I_c$ of the first category 
subsets of $\reals^n$.

\corollary{cor:mainA}{ If GCH holds 
then there is a Hausdorff, connected and locally connected topology 
$\tau_{\cal C}$ on $\reals^m$ such that for any family 
$\F\subset\C(\T_{\cal O}^m,\T_{\cal O}^n)$, 
of ordinary continuous functions,
containing all constant functions 
and such that ${\rm int}_{\tau_{\cal O}}(\{x\in X\colon f(x)=g(x)\})=\emptyset$
for every distinct $f,g\in\F$
we have
\[
\F=\C(\tau_{\cal F},\tau_{\cal C}),
\]
where $\tau_{\cal F}$ is generated
by the family $\{f^{-1}(U)\colon U\in\tau_{\cal C}, f\in\F\}$.
Moreover, $\tau_{\cal F}$ is connected
and locally connected, and it is Hausdorff provided \F\ separates points. 
It is also
consistent with ZFC+GCH that all these topologies are 
completely regular and Baire.
}

In particular, it can be shown 
that for any different harmonic functions
$f,g\colon \reals^n\to\reals^m$ we have 
${\rm int}_{\tau_{\cal O}}(\{x\in X\colon f(x)=g(x)\})=\emptyset$.\footnote{
To see it let $f,g\colon \smallreals^n\to\smallreals$ be two 
harmonic functions that agree on some neighbourhood of $x_0\in\smallreals^n$.
Let $R>0$ be a supremum of all balls 
$B[x_0,r]=\{x\in\smallreals^n |x-x_0|\leq r\}$ on which $f$ and $g$ agree.
If $R=\infty$ then we are done. So, assume that $R<\infty$.
In particular, $f$ and $g$ agree on $B[x_0,R]$. Now, for every point 
$x$ in a boundary of $B[x_0,R]$ use Cauchy-Kovaleski Theorem
\cite[Theorem 1.5 page 330]{ChazPiri:DifEq}
for Laplacian operator $L$ with initial values given by the
derivatives of $f$ on a boundary of $B[x_0,R]$
to find an open neighbourhood $U_x$ of $x$ in which 
this initial value problem has a unique solution. Since both functions
$f$ and $g$ form the solutions for this initial value problem 
(they agree on $B[x_0,R]$, so their derivatives must also agree on the 
boundary of $B[x_0,R]$) they must be equal on $U_x$. Now, 
$\{U_x\}_{|x-x_0|=R}$ is an open cover of a compact set
$\{x\colon |x-x_0|=R\}$ so, we can find finite subcover of it. 
But this means that we can find $r>R$ such that $f$ and $g$ 
agree on $B[x_0,r]$, contradicting maximality of $R$.

Notice also, that this scheme can be used to any class of functions defined
by operator for which we can use Cauchy-Kovaleski Theorem.} 
Thus, by Corollary \ref{cor:mainA}, the class of all harmonic functions
$f\colon \reals^n\to\reals^m$ can be topologized.

Another $\sigma$-ideal that
can be used with Theorem \ref{main} 
is the ideal $\I_\omega$ of at most countable sets. 
Since for any two different 
analytic functions $f,g\in\cal A$ we have 
$\{x\colon f(x)=g(x)\}\in\I_\omega$, we can also conclude the
following corollary. 

\corollary{cor:mainB}{ If GCH holds 
then there is a Hausdorff, connected and locally connected topology 
$\tau_{\cal A}$ (on $\reals$ or $\ \complexnumb$) such that for any family 
$\F\subset\A$ containing all constant functions 
we have
\[
\F=\C(\tau_{\cal F},\tau_{\cal A}),
\]
where $\tau_{\cal F}$ is generated
by the family $\{f^{-1}(U)\colon U\in\tau_{\cal A}, f\in\F\}$.
Moreover, $\tau_{\cal F}$ is connected
and locally connected, and it is Hausdorff provided \F\ separates points. 
It is also
consistent with ZFC+GCH that all these topologies are 
completely regular and Baire.
}

Notice also, that if the family \F\ 
in Corollary \ref{cor:mainB} is closed under the composition and
$id\in\F$, then, by Theorem \ref{th:properties}(v),
$\F=\C(\tau_{\cal F})$. We can write this in the form of next corollary.

\corollary{cor:mainC}{ If GCH holds  and \F\ is a 
family of real 
functions which is closed under the composition and such that 
$\{id\}\cup\const\subset\F\subset\A$, then there exists a Hausdorff, 
connected and locally connected topology $\T_{\cal F}$ 
(on $\reals$ or \complexnumb) such that
$\F=\C(\T_{\cal F})$. In particular, there exist a 
\underline{``linear topology''}
$\T_{\cal L}$,
a \underline{``polynomial topology''} $\T_{\cal P}$
and an \underline{``analytic topology''} 
$\T_{\cal A}$ which are Hausdorff, connected
and locally connected such that 
$\T_{\cal L}\subset\T_{\cal P}\subset\T_{\cal A}$
and for which \underline{$\Lin=\C(\T_{\cal L})$, 
$\cal P=\C(\T_{\cal P})$ and $\A=\C(\T_{\cal A})$.}
Moreover, it is 
consistent with ZFC+GCH that all these topologies are 
completely regular and Baire.
}

The three corollaries above show that ``nice'' 
classes of real functions can be ``nicely''
topologized. However, Theorem \ref{main} 
tells us also that 
variety of ``wild'' classes
of real functions can be topologized as well. 
This is the case, for example, for the families
$\F_1=\const\cup\{x^3,e^x\}$, 
$\F_2=\const\cup\{x^5 - 17,\sin x,1/(x^2+1)\}$
and $\F_3=\const\cup\{x,\ln(x^2+2),g\}$, 
where $g(x)=e^{-x^{-2}}$ for $x\neq 0$, $g(0)=0$, is 
well known
\cinfinity\ function which is not analytic. Also, 
the functions in the class \F\ must be neither 
measurable nor have the Baire
property, since in Corollaries \ref{cor:mainA} and \ref{cor:mainB}
the families $\C(\ordinarytop)$  and $\cal A$ can be replaced by 
any family $\R_0$ of real functions as long as
$|\R_0|\leq\continuum$. In particular,

\corollary{cor:mainD}{ If GCH holds  and 
$h\colon\reals\to\reals$ is any one-to-one function then the family
$\const\cup\{h\}$ can be topologized.}

The next corollary gives a negative answer for the following question of
Lee Larson (private communication): ``Let $\tau_1$ and $\tau_2$ 
be homothetically closed 
connected topologies on \reals. Is it true that either
$\C(\tau_1,\tau_2)=\const$
or $\Lin\subset\C(\tau_1,\tau_2)$?''

\corollary{cor:Lee}{ If GCH holds 
then there exist homothetically closed 
Hausdorff connected topologies $\tau_1$ and $\tau_2$ on \reals\ such that
$\C(\tau_1,\tau_2)\neq\const$ while $\C(\tau_1,\tau_2)\cap\Lin=\const$.}

Proof. Let $f(x)=x^3$ for all $x\in\reals$ and let 
$\F=\{L\circ f\circ M\colon L,M\in\Lin\}\subset\A$. Then, 
by Corollary \ref{cor:mainB},
\F\ can be topologized as $\C(\tau_1,\tau_2)$.
Moreover, by Theorem \ref{th:properties}(iv), the 
topology $\tau_2$
can be taken as homothetically closed. It is also easy to see that 
if $\tau_1$ is a week topology, as in Theorem \ref{main}, then it is
 homothetically closed, connected and Hausdorff.
This finishes the proof.

\bigskip

In fact, the assumption GCH in 
Corollary \ref{cor:Lee}
is unnecessary. It will follow from the Theorem \ref{mainZFC}.

\section{$\!\!\!\!\!\!\!${\bf.} Proof of Theorem 4(A). }\label{SecA}

In what follows we will write $H(A,B)$ for the set of all functions
from a finite subset of $A$ into $B$.

Take $X,Y,\I,\R$ as in Theorem \ref{main}.
In both parts of the 
theorem the topology $\tau_2$ will be chosen in the following way.
We will choose a topological space $S$, and 
construct a one-to-one function
$e\colon Y\to S^{(2^\omega)^+}$.
Topology $\tau_2$ on $Y$ will be defined as a 
weak topology generated by the function
$e\colon Y\to S^{(2^\omega)^+}$. Thus, 
$Y$ will be identified with the
subspace $e[Y]$ of $S^{(2^\omega)^+}$. 

Notice that if $\B_0\subset{\cal P}(S)$ is such that 
$\B_0\cup\{S\}$ forms a basis for $S$ 
then the sets 
\[
[\delta]_Y=\{y\in Y\colon (\forall d\in{\rm dom}(\delta))
(e(y)(d)\in\delta(d)\},
\]
with $\delta\in H((\continuum)^+,\B_0)$, 
form a basis for $(Y,\tau_2)$. 

Now, if $\F$ is as in Theorem \ref{main} and we define 
$e_1\colon X\to S^{{\cal F}\times{(2^\omega)^+}}$ by formula
$e_1(x)(f,\xi)=e(f(x))(\xi)$ then the sets 
\[
[\e]_X=\{x\in X\colon (\forall d\in{\rm dom}(\e))(e_1(x)(d)\in\e(d)\},
\]
with $\e\in H({\cal F}\times (2^\omega)^+,\B_0)$,
will form a basis for the weak topology $\tau$ on $X$ generated by $\F$, 
since for $f\in\F$ and $\{<\xi,B>\}\in H((\continuum)^+,\B_0)$
\begin{eqnarray*}
f^{-1}([<\xi,B>]_Y) 
   & = & \{x\colon f(x)\in[<\xi,B>]_Y\} \\
   & = & \{x\colon e_1(x)(f,\xi)=e(f(x))(\xi)\in B\}\\
   & = & \{x\colon x\in[<<f,\xi>,B>]_X\}\\
   & = & [<<f,\xi>,B>]_X.
\end{eqnarray*}
Thus, $X$ can be ``identified'' with $e_1[X]$. 
(Notice that the ``identifying'' function 
$e_1$ does not have to be one-to-one.) 

It is easy to see that for such topologies we have
$\F\subset\C(\tau,\tau_2)$.
Thus, the problem in our construction
will be to show that any function $f\in Y^X\setminus\F$
is not in $\C(\tau,\tau_2)$. This will be done 
by choosing an appropriate space $S$ and an embedding $e$.
Function $e$ will be naturally identified with the mapping
from $Y\times (2^\omega)^+$ into $S$.

In the case of the proof of Theorem \ref{main}(A)
we will choose $S$ to be 
the space $P=\{0,1,2\}$ with topology
$\{\emptyset,P,\{0\},\{1\},\{0,1\}\}$ and $\B_0=\{\{0\},\{1\}\}$.
The construction of $e$ will be done by induction on 
$\xi<(\continuum)^+$
by listing all functions from $Y^X$ as a sequence 
$<h_\zeta\colon\zeta\in(2^\omega)^+>$, constructing an increasing
sequence of functions $e|_{Y\times\xi}$ and, in step $\xi\in(2^\omega)^+$,
defining $e|_{Y\times\{\xi\}}$ in such a way that 
$h_\xi^{-1}([<\xi,\{0\}>]_Y)\not\in\tau$ provided 
$h_\xi\not\in\C(\tau,\tau_2)$.

\bigskip

We will need two technical lemmas for our constructions.
The first one will be used in both parts of the proof of Theorem \ref{main}.
More precisely, assumptions (A) and (B) of Lemma \ref{lem:aaa}
correspond to the proofs of parts (A) and (B) of Theorem \ref{main}, 
respectively. 
(Thus, if the reader likes to skip the proof of Theorem \ref{main}(B), he
may skip the proof of second part of Lemma \ref{lem:aaa} as well.) 

\lemma{lem:aaa}{Let $(X,\tau_1)$ and $(Y,\tau_2)$ be 
connected topological spaces of cardinality 
\continuum\ such that $Y$ is Hausdorff and that
every $A\in[Y]^{<2^\omega}$ is closed in $Y$.
Moreover, let $\const\subset\F\subset \C(\tau_1,\tau_2)$
and let $\J_0$ be a $\sigma$-ideal generated by 
all sets
$\{x\colon f(x)=g(x)\}$ for $f,g\in\F$, $f\neq g$.
If \J\ is a $\sigma$-ideal containing $\J_0$ and \B\ is a base for $X$ 
such that 
\begin{equation}\label{ab}
U\setminus D \mbox{ is nonempty and connected for every } 
U\in\B \mbox{ and } D\in\J 
\end{equation}
then for every
$h\in\C(\tau_1,\tau_2)\setminus\F$ there is $x\in X$ such that for every 
$U\in\tau_1$ with $x\in U$, $D\in\J$
and $\F_0\in[\F]^{\leq\omega}$ the following is true.
If either

\begin{description}

\item[(A)] $\F_0$ is finite; or

\item[(B)] $\J=\J_0$, $X$ is a regular space and 
           $\bigcap_{n<\omega}U_n\neq\emptyset$
           for every sequence $\{U_n\in\B\colon n<\omega\}$ 
           such that ${\rm cl}(U_{n+1})\subset U_n$,

\end{description}
then 
\begin{equation}\label{abc}
|h(\{z\in U\setminus D\colon h(z)\neq f(z) 
                  \mbox{ for all } f\in\F_0\})|=\continuum.
\end{equation}
}

Proof. Let $h\in\C(\tau_1,\tau_2)\setminus\F$.

First notice that there is $x\in X$ such that for every 
$U\in\tau_1$ with $x\in U$,
\begin{equation}\label{restriction}
h_{|U}\neq f_{|U} \mbox{ for all } f\in\F.
\end{equation}

To see this, assume, by way of contradiction, that for every 
$z\in X$ there
is $f_z\in\F$ and $U_z\in\tau_1$ with $z\in U_z$, such that 
$h_{|U_z}=(f_z)_{|U_z}$. Let $U\in\tau_1$ be a maximal, nonempty set such that
$h_{|U}=f_{|U}$ for some $f\in\F$. If $U\neq X$ then, by connectedness
of $(X,\tau_1)$, there is 
$z\in\mbox{cl}_{\tau_1}(U)\setminus U$. 
Then $U_z\cap U\neq\emptyset$ and  
$(f_z)_{|(U_z\cap U)}=h_{|(U_z\cap U)}=f_{|(U_z\cap U)}$, i.e.,
$f$ and $f_z$ are equal on a nonempty open set. Thus, by (\ref{ab}),
$f_z=f$, contradicting maximality of $U$. Hence, $U=X$. But then, 
$h=h_{|U}=f_{|U}=f$ contradicting the fact that $h\not\in\F$.
The condition (\ref{restriction}) is proved.

Now, choose $x$ as in (\ref{restriction}).
We will show that Lemma \ref{lem:aaa} holds for $h$ and $x$.

For $f\in\F$ let 
\[
P_f=\{z\in X\colon h(z)=f(z)\}.
\]
Thus, (\ref{restriction}) tells us that
\begin{equation}\label{restrictionA}
U\not\subset P_f\ \mbox{ for all }\ f\in\F
\ \mbox{ and }\ U\in\tau_1
\ \mbox{ with }\ x\in U.
\end{equation}
Assume, by way of contradiction, that there are $W\in\tau_1$ with
$x\in W$, $D_0\in\J$ and $\F_0\in[\F]^{\leq\omega}$ for which
condition (\ref{abc}) does not hold, i.e., 
such that $|h(B\setminus D_0)|<\continuum$ where 
\[
B=\{z\in W\colon h(z)\neq f(z)\mbox{ for all }f\in\F_0\}
= W\setminus\bigcup_{f\in{\cal F}_0}P_f.
\] 
We will show that this contradicts (A) and (B).

Without loss of generality we may assume that the constant function equal
to $h(x)$ is in $\F_0$. Hence, $h(x)\not\in h(B)$.
Since $h(B\setminus D_0)$ is closed in $\tau_2$,
$U=W\setminus h^{-1}(h(B\setminus D_0))\in\tau_1$ and
$x\in U\subset W\setminus (B\setminus D_0)$. Decreasing $U$, if necessary,
we may assume also that $U\in\B$.
Since $U\subset W$ and
\[
(U\setminus D_0)\cap\left(W\setminus\bigcup_{f\in{\cal F}_0}P_f\right)=
(U\setminus D_0)\cap B=U\cap (B\setminus D_0)=\emptyset
\]
then
\begin{equation}\label{cond_PPP}
U\setminus D_0\subset\bigcup_{f\in{\cal F}_0}P_f.
\end{equation}
Notice also that 
the sets $P_f$ are closed, since $\tau_2$ is Hausdorff, and that
for every $f\in\F$ and $D\in\J$,
\begin{equation}\label{cond_UD}
U\subset P_f \mbox{ if and only if } U\setminus D\subset P_f.
\end{equation}
Condition (\ref{cond_UD}) follows from the fact that, 
by (\ref{ab}), $U\setminus D$
is dense in $U$ and that $U\setminus D\subset P_f$ implies
$U\subset {\rm cl}(U\setminus D)\subset P_f$.

Notice also that (\ref{restrictionA}) and (\ref{cond_UD}) in particular
imply that 
\begin{equation}\label{con10}
U\setminus D\not\subset P_f \mbox{ for every } f\in\F \mbox{ and } D\in\J
\end{equation}
so that, by (\ref{cond_PPP}), $|\F_0|>1$.


We have two cases to consider.

{\bf Case (A).} Decreasing $\F_0$, if necessary, we may assume that
$\F_0$ is minimal family satisfying (\ref{cond_PPP}), i.e., 
that we have also
\begin{equation}\label{cond_P}
P_f\cap(U\setminus D_0)\not\subset\bigcup_{g\in{\cal F}_0\setminus\{f\}}P_g
\mbox{ for every } f\in\F_0. 
\end{equation}
Put 
$D_1=\bigcup\{P_f\cap P_g\colon f,g\in\F_0,f\neq g\}\subset 
\{z\colon f(z)=g(z)\mbox{ for some }f,g\in\F_0,f\neq g\}\in\J$
and $D=D_0\cup D_1\in\J$. 
Then, $U\setminus D$ is connected in $\tau_1$ and
the family $\{P_f\cap(U\setminus D)\}_{f\in{\cal F}_0}$
forms a partition of $U\setminus D$ by the sets relatively closed in 
$U\setminus D$. Moreover, $\F_0$ has 
at least two elements and,
by (\ref{cond_P}), all these sets must be nonempty
since $D_1\subset\bigcup_{g\in{\cal F}_1\setminus\{f\}}P_g$
and $P_f\cap(U\setminus D)=(P_f\cap (U\setminus D_0))\setminus D_1$.
This contradicts connectedness of $U\setminus D$.
Case (A) is completed.

{\bf Case (B).} Notice that, by (\ref{ab}), 
${\rm int}(P_f)\cap{\rm int}(P_g)=\emptyset$
for every $f,g\in\F$, $f\neq g$.
Replacing sets $P_f$ with sets 
$P_f\setminus\bigcup\{{\rm int}(P_g)\colon g\in\F_0, g\neq f\}$, 
if necessary, we can assume that in addition to (\ref{cond_PPP}) and
(\ref{con10}) we have
\begin{equation}\label{con22}
P_g\cap{\rm int}(P_f)=\emptyset \mbox{ for all } f,g\in\F_0, g\neq f.
\end{equation}
Enumerate $\F_0$ as $\{f_n\colon 0<n<\omega\}$.
Increasing $D_0$, if necessary, we can also assume that
$D_0=\bigcup_{0<n<\omega}D_n$ where sets $D_n$, $0<n<\omega$, 
are of the form
$\{x\colon f(x)=g(x)\}$ for $f,g\in\F$, $f\neq g$, i.e., $D_n$ are
closed and nowhere dense.
We will construct a sequence 
$\{U_n\in\B\colon n<\omega\}$ 
of open subsets of $U$ such that for every $n<\omega$ we will have
\[
{\rm cl}(U_{n+1})\subset U_n, \,\,\,\,
U_{n+1}\cap(P_{f_{n+1}}\cup D_{n+1})=\emptyset \mbox{ and }
U_n\not\subset P_f \mbox{ for every } f\in\F.
\]
This will finish the proof, since, by (B), we will get
$\emptyset\neq\bigcap_{n<\omega}U_n\subset 
(U\setminus D_0)\setminus \bigcup_{f\in{\cal F}_0}P_f$ 
contradicting (\ref{cond_PPP}).

To see that the construction is possible start with putting $U_0=U$.
It satisfies the inductive hypothesis by (\ref{con10}).
So, assume that $U_n$ 
is already constructed for some $n<\omega$.
If $P_g\cap(U_n\setminus D_0)\subset P_{f_{n+1}}$ for every $g\in\F_0$
then, by (\ref{cond_PPP}), $U_n\setminus D_0\subset P_{f_{n+1}}$
and, by (\ref{cond_UD}), $U_n\subset P_{f_{n+1}}$ contradicting
our inductive assumption. So, 
choose $g\in\F_0$ such that 
$\emptyset\neq P_g\cap(U_n\setminus D)\not\subset P_{f_{n+1}}$, put
$D^{\prime\prime}=P_g\cap P_{f_{n+1}}\subset\{z\colon g(z)=f_{n+1}(z)\}\in\J$
and $D^\prime=D_0\cup D^{\prime\prime}\in\J$.
Then, 
$P_g\cap(U_n\setminus D^\prime)=
(P_g\cap(U_n\setminus D_0))\setminus P_{f_{n+1}}
\neq\emptyset$ and
$P_g\cap(U_n\setminus D^\prime)\neq U_n\setminus D^\prime$ since 
the equation 
$P_g\cap(U_n\setminus D^\prime)=U_n\setminus D^\prime=
(U_n\setminus D_0)\setminus D^{\prime\prime}$ would imply
$U_n\setminus D_0\subset P_g\cup D^{\prime\prime}\subset P_g$
and, by (\ref{cond_UD}), $U_n\subset P_g$, 
contradicting our inductive assumption.
Thus, $P_g\cap(U_n\setminus D^\prime)$
cannot be open in $U_n\setminus D^\prime$ since it is closed 
in $U_n\setminus D^\prime$ and
$U_n\setminus D^\prime$ is connected.
So, choose $z\in(U_n\setminus D^\prime)\cap(P_g\setminus{\rm int}(P_g))$.
Since $z\not\in P_{f_{n+1}}\cup D_{n+1}$ and $X$
is regular, we can choose $U_{n+1}$ such that
$z\in U_{n+1}\subset {\rm cl}(U_{n+1})\subset U_n$ and
$U_{n+1}\cap (P_{f_{n+1}}\cup D_{n+1})=\emptyset$. To finish the proof it is 
enough to show that $U_{n+1}\not\subset P_f$ for every $f\in\F$.
So, by way of contradiction assume that
there is $f\in\F$ such that $U_{n+1}\subset P_f$.
Since $z\not\in{\rm int}(P_g)$ we have $U_{n+1}\not\subset P_g$
and so, $f\neq g$. But then, $z\in P_g\cap U_{n+1}\subset{\rm int}(P_f)$
contradicting (\ref{con22}).
Lemma \ref{lem:aaa} has been proved.

\bigskip

We already noticed that topologies 
$\tau$ and $\tau_2$ should be connected.
The next lemma explains how we are going to achieve this goal
for the proof of Theorem \ref{main}(A). In the 
next lemma $[\delta]$ will stand
for 
\[
[\delta]=\{g\in P^Z\colon\delta\subset g\}.
\]

\lemma{lem:connected}{ Let $Z$ be an arbitrary set and let
$M\subset P^Z$ be such that 
$[\delta]\cap M\neq\emptyset$ for every $\delta\in H(Z,3)$.
Then for every $\delta\in H(Z,2)$
the set $[\delta]\cap M$ is connected in $P^Z$.
In particular, $M$ considered as a subspace of $P^Z$
is connected and locally connected.}

Proof. For the use of this proof let $[\delta]$ denote
$[\delta]\cap M$ for $\delta\in H(Z,3)$.
Then, for every 
$\varepsilon,\delta\in H(Z,3)$
\[
[\varepsilon]\cap[\delta]\neq\emptyset
\mbox{ if and only if }
\varepsilon\cup\delta\in H(Z,3)
\]
and
\begin{equation}\label{cond_inclusion}
[\varepsilon]\subset[\delta]
\mbox{ if and only if } \delta\subset\varepsilon.
\end{equation}
To argue for (\ref{cond_inclusion}) it is enough
to show that $[\varepsilon]\subset[\delta]$ 
implies $\delta\subset\varepsilon$,
since the other inclusion is obvious.
But if $\delta\not\subset\varepsilon$ then there exists
$\varepsilon^\prime\in H(Z,3)$ extending $\varepsilon$ such that
$\varepsilon^\prime\cup\delta\not\in H(Z,3)$.  Hence, 
$[\varepsilon^\prime]\cap[\delta]=\emptyset$, while 
$\emptyset\neq[\varepsilon^\prime]
\subset[\varepsilon]$. This contradicts 
$[\varepsilon]\subset[\delta]$.

Now, let us turn to the proof of connectedness of 
$[\delta]$, where $\delta\in H(Z,2)$.
Let $\varepsilon_0,\varepsilon_1\in H(Z,2)$ 
be such that $[\varepsilon_0]$ and 
$[\varepsilon_1]$ are
nonempty disjoint subsets of $[\delta]$.
Then, by (\ref{cond_inclusion}),
$\delta\subset\varepsilon_i$ for $i<2$, i.e., 
$\varepsilon_i=\delta\cup\delta_i$ for some 
$\delta_i\in H(Z,2)$ such that 
$\mbox{dom}(\delta)\cap\mbox{dom}(\delta_i)=\emptyset$.
Let $D=\mbox{dom}(\delta_0)\cup\mbox{dom}(\delta_1)$ and define
$\eta\colon D\to 3$ by $\eta(d)=2$ for all $d\in D$.
Then, $\emptyset\ne[\eta\cup\delta]\subset[\delta]$.
We will see that 
$[\eta\cup\delta]\subset{\rm cl}[\varepsilon_0]
\cap{\rm cl}[\varepsilon_1]$. This will clearly imply that 
$[\delta]$ is connected.

So, let
$\xi\in H(Z,2)$ be such that $[\xi]\cap[\eta\cup\delta]\ne\emptyset$.
Then, $\xi\cup\eta\cup\delta\in H(Z,3)$, i.e, for $i<2$,
$\xi\cup\varepsilon_i=\xi\cup\delta_i\cup\delta\in H(Z,2)$
and so,
$[\xi]\cap[\varepsilon_i]\neq\emptyset$. But this means that 
the set $[\eta\cup\delta]\subset[\delta]$ 
is in the closure of
both $[\varepsilon_0]$ and $[\varepsilon_1]$. 

Since sets $[\delta]=[\delta]\cap M$ for $\delta\in H(Z,2)$ form basis for
$M$ we conclude that indeed $M$ is connected and locally connected.
Lemma \ref{lem:connected} is proved.

\bigskip

Now we are ready for the main part of the proof.

\bigskip

{\bf Construction of embedding e.} Assume GCH and take
$X,Y,\I$ and $\R$ as in Theorem \ref{main}. 
By the structure of condition (\ref{conThmain}) it is easy to see that 
we can assume that $\R\cap\const=\emptyset$ and
\[
f^{-1}(y)\in\I \mbox{ for all } f\in\R \mbox{ and } y\in Y. 
\]
Let $\I_0$ be a family of all $I\in\I$ such that either $|I|=1$ or
$I=\{x\in X\colon f(x)=g(x)\}\in\I$ for some $f,g\in\R$
and let $\J\subset\I$ be the 
$\sigma$-ideal generated by $\I_0$.
Let $\B_1=\B_0\cup\{\{2\}\}=\{\{0\},\{1\},\{2\}\}$ and
for $T\subset(\continuum)^+$ define 
$\H(T)$ as the set of all 
$\varepsilon\in H(\R\times T,\B_1)$ such that
\[
\{x\colon f(x)=g(x)\}\in\I \mbox{ for all } 
<f,\eta>,<g,\eta>\in{\rm dom}(\varepsilon), f\neq g.
\]
Moreover, let
\begin{equation}\label{condGCH}
\{<h_\zeta,x_\zeta,A_\zeta>\colon\zeta<(\continuum)^+\}
\end{equation}
be an enumeration of 
$Y^X\times X\times[Y]^{<2^\omega}$.
This can be chosen by GCH. 

We will construct, by induction on $\zeta<(\continuum)^+$,
an increasing sequence of functions 
$\{e|_{Y\times\zeta}\colon \zeta<(\continuum)^+\}$ that
will satisfy the following inductive conditions for all
$\eta<\zeta<(\continuum)^+$, when we adopt the notation introduced
on the beginning of this section:
\begin{description}

\item[(a)]   $A_\eta\subset[\{<\eta,\{1\}>\}]_Y$ and 
             $h_\eta(x_\eta)\in[\{<\eta,\{0\}>\}]_Y$ provided 
             $h_\eta(x_\eta)\not\in A_\eta$;

\item[(b)]   $|[\delta]_Y|=\continuum$ for all 
             $\delta\in H(\zeta,\B_1)$;

\item[(c)]   $[\varepsilon]_X\not\in\J$
             for all $\varepsilon\in\H(\zeta)$;

\item[(d)]   $[\varepsilon]_X\cap h_\eta^{-1}([\{<\eta,\{2\}>\}]_Y)\not\in\J$
             for all $\varepsilon\in\H(\zeta)$ such that for every $I\in\J$
             \begin{equation}\label{can_d}
             |h_\eta(\{z\in[\varepsilon]_X\setminus I
             \colon h_\eta(z)\neq f(z) \mbox{ for all } 
             <f,\eta>\in{\rm dom}(\varepsilon)\})|=\continuum.
             \end{equation}
\end{description}

First, let us see how conditions (a)-(c) imply the theorem.
So, let $\F$ be as in the theorem and let topology
$\tau_2$ be chosen as described in the beginning of the section.
Condition (a) clearly implies that every $A\in[Y]^{<2^\omega}$ 
can be separated from every point $y\in Y\setminus A$. 
(Simply choose $\eta\in (\continuum)^+$ such that $A_\eta=A$,
$h_\eta\equiv y$.) Thus, $Y$ is Hausdorff and every 
$A\in[Y]^{<2^\omega}$ is closed in $Y$. 
This also implies that
$e$ is one-to-one and that $(X,\tau)$ is Hausdorff, provided
$\F$ separates points. 

Condition (b) and 
Lemma \ref{lem:connected} used with $M=e[Y]$ 
imply that $(Y,\tau_2)$ is connected and locally connected,
since $M\cap[\{<\xi,i>\}]=e[[\{<\xi,\{i\}>\}]_Y]$.
Similarly, since 
$H(\F\times(\continuum)^+,\B_1)\subset\H((\continuum)^+)$
we can use (c) and 
Lemma \ref{lem:connected} with $M=e_1[X\setminus D]$, $D\in\J$,  
to conclude that $(X,\tau)$ is
connected, locally connected and that 
$[\varepsilon]_X\setminus D$ is connected for every $D\in\J$
and every $\varepsilon\in H(\F\times(\continuum)^+,\B_0)$.
Therefore, 
condition (\ref{ab}) from Lemma~\ref{lem:aaa} is satisfied
with family $\B$ defined by 
$\B=\{[\e]_X\colon \e\in H(\F\times(\continuum)^+,\B_0)\}$ and $\J$ defined
as above.
To finish the argument it is enough to show that
$\C(\tau,\tau_2)\subset\F$, since the converse inclusion is clear.
By way of contradiction assume that there is 
$h\in\C(\tau,\tau_2)\setminus\F$. 
Let $x$ be as in Lemma~\ref{lem:aaa}
and let $\eta\in (\continuum)^+$ be such that $h=h_\eta$, $x=x_\eta$ and 
$A_\eta=\emptyset$. Then, by (a), $x\in h^{-1}([\{<\eta,\{0\}>\}]_Y)$. 
Moreover, by  Lemma~\ref{lem:aaa},
condition (\ref{can_d}) is satisfied
for every $[\varepsilon]_X\in\B$ such that $x\in[\varepsilon]_X$.
Thus, by (d), 
$[\varepsilon]_X\cap h^{-1}([\{<\eta,\{2\}>\}]_Y)\neq\emptyset$.
Hence, none basic open set $[\varepsilon]_X\in\tau$
with $x\in[\varepsilon]_X$,
can be contained in $h^{-1}([\{<\eta,\{0\}>\}]_Y)\neq\emptyset$
and so, $h$ cannot be continuous. This contradiction shows that
$\F=\C(\tau,\tau_2)$.

\medskip

To finish the proof it is enough to make the inductive
construction. Let $\{I_\xi\colon \xi<\continuum\}$ be an enumeration of
$\I_0$ and 
let us assume that for some $\zeta<(\continuum)^+$ 
the construction is indeed done. We will construct 
$e|_{Y\times\{\zeta\}}$. To do this, let
\[
\{<\delta^0_\xi,\varepsilon^0_\xi,\alpha_\xi,\delta_\xi,\varepsilon_\xi>
\colon 0<\xi<\continuum\}
\]
be an enumeration of 
$H(\zeta,\B_1)\times\H(\zeta)\times(\zeta+1)\times 
H(\{\zeta\},\B_1)\times\H(\{\zeta\})$ 
with each tuple appearing
in the sequence continuum many times. We will construct, by
induction on $\xi<\continuum$, an increasing sequence of functions
$\{e|_{Y_\xi\times\{\zeta\}}\}$, where $Y_\xi\subset Y$, 
by starting with $Y_0=A_\zeta\cup\{h_\zeta(x_\zeta)\}$,
defining $e(d,\zeta)=1$ for $d\in A_\zeta$, $e(h_\zeta(x_\zeta),\zeta)=0$
if $h_\zeta(x_\zeta)\not\in A_\zeta$ and
such that the following inductive conditions
are hold for $0<\xi<\continuum$:
\begin{description}
 
\item[(i)]   $Y_\xi\setminus\bigcup_{\eta<\xi}Y_\eta$ is finite;

\item[(ii)]  there is $y\in Y_\xi\setminus\bigcup_{\eta<\xi}Y_\eta$ such that
             $y\in[\delta^0_\xi]_Y\cap[\delta_\xi]_Y$;

\item[(iii)] there exists 
             $x\in[\varepsilon^0_\xi]_X\setminus\bigcup_{\eta<\xi}I_\eta$
             such that 
  $f(x)\in\varepsilon_\xi(f,\zeta)\cap Y_\xi\setminus\bigcup_{\eta<\xi}Y_\eta$
             for all \mbox{$<f,\zeta>\in{\rm dom}(\varepsilon_\xi)$};
             moreover, we have
             $h_{\alpha_\xi}(x)\in [\{<\alpha_\xi,\{2\}>\}]_Y$
             provided for every $I\in\J$
\end{description}
             \[
             |h_{\alpha_\xi}(\{z\in[\e^0_\xi]_X
             \setminus I\colon h_{\alpha_\xi}(z)\neq f(z) \mbox{ for all } 
             <f,\zeta>\in{\rm dom}(\varepsilon_\xi)\})|=\continuum.
             \]


Notice that this will imply (a)-(d)
when we extend $\bigcup_{\xi<2^\omega} e|_{Y_\xi\times\{\zeta\}}$
to $e|_{Y\times\{\zeta\}}$
arbitrarily. Clearly (a) is implied by the definition of $e$ on 
$Y_0\times\{\zeta\}$.
Condition (b) is implied by (ii), since functions 
$\delta^0_\xi\cup\delta_\xi$ list all $H(\zeta+1,\B_1)$ and each function
appears there continuum many times.
Finally, (c) and (d) are implied by (iii) in similar way,
if we notice that every function from $\H(\zeta+1)$ appears as
$\varepsilon^0_\xi\cup\varepsilon_\xi$ for continuum many $\xi$ and
that for every $I\in\J$ there is $\xi<\continuum$ such that 
$I\subset\bigcup_{\eta<\xi}I_\eta$.

So, let us assume that for some $\xi<\continuum$ construction is done.
Notice that the set $\bigcup_{\eta<\xi}Y_\eta$ has cardinality less
that continuum.
To get $y$ satisfying (ii) it is enough to choose 
$y\in[\delta^0_\xi]_Y\setminus\bigcup_{\eta<\xi}Y_\eta$ 
and define $e(y,\zeta)=i$ where $\delta_\xi=\{<\zeta,\{i\}>\}$.
To get (iii) let
$\varepsilon_\xi=\{<f_j,\zeta,\{i_j\}>\colon j<n\}$ and put 
\[
I=\{x\in X\colon f_j(x)=f_k(x) \mbox{ for some } j<k<n\}.
\]
Then $I\in\I$, since $\varepsilon_\xi\in\H(\{\zeta\})$.
The set $T_0=\{y\}\cup\bigcup_{\eta<\xi}Y_\eta$ 
has cardinality $<\continuum$ so 
$T=I\cup\bigcup_{j<n}f_j^{-1}(T_0)\in\J$
and, by the inductive hypothesis, we can choose 
$x\in[\varepsilon^0_\xi]_X\setminus(T\cup\bigcup_{\eta<\xi}I_\eta)$.
Since $f_j(x)\neq f_k(x)$ 
for $j<k<n$ and $f_j(x)\not\in T_0$ 
we can define $e(f_j(x),\zeta)=i_j$ for $j<n$. 
This gives the main part of (iii).
Moreover, if additional assumption is satisfied than 
we can choose 
$x\in
\{z\in[\e^0_\xi]_X\setminus(T\cup\bigcup_{\eta<\xi}I_\eta)
\colon h_{\alpha_\xi}(z)\neq f_j(z) \mbox{ for all } j<n\}$
such that $h_{\alpha_\xi}(x)\not\in T_0$. So, we can freely define
$e(h_{\alpha_\xi}(x),\zeta)=2$. The construction
is finished if we define
$Y_\xi=\{f_j(x)\colon j<n\}
\cup\{h_{\alpha_\xi}(x),y\}\cup\bigcup_{\eta<\xi}Y_\eta$.
(We define $e(h_{\alpha_\xi}(x),\zeta)$ arbitrarily, when the additional
requirement of (iii) is not satisfied.)

This finishes the proof of Theorem \ref{main}(A).

\section{$\!\!\!\!\!\!\!${\bf.} 
Discussion of the assumptions and 
generalizations of Theorem 4. }\label{SecDisc}

We start here with noticing that all families \F\ from 
Theorem \ref{main} can be topologized
with the same topology $\tau_2$ on the range. 
It would be nice to prove Theorem \ref{main} with $\R=Y^X$, i.e., to have 
the same universal topology $\tau_2$ 
that could be used for topologizing
all families $\F\subset Y^X$ containing constant functions  
and satisfying condition (\ref{conThmain})
from Theorem \ref{main}. However, this
cannot be done at least as long 
as we assume that $X=Y$ and that 
the ``universal'' topology $\tau_2$ contains a set $U$ 
such that $|U|=|X\setminus U|=\continuum$.
This is the case, since then for a bijection
$f\colon X\to X$ such that 
$f(X\setminus U)=U$ and $f(x)\neq x$ for all $x\in X$
the family $\const\cup\{id_X,f\}$ could be topologized. 
But the family $\const\cup\{id_X,f\}$ cannot be topologized since 
the domain
topology would have to contain both $U$ and $X\setminus U$, 
and thus, the class
of continuous functions would contain also a characteristic 
function of $U$.

\bigskip

It is not clear at this point whether the Theorem \ref{main}(A) or (B) 
can be proved without any additional set theoretical assumptions.
However, it is easy to see that the real assumptions we have used
in the proof is that $2^{2^\omega}=(\continuum)^+$ and that 
$\bigcup\G\neq X$ for every $\G\in[\I]^{< 2^\omega}$.
For general ideals, this last assumption does 
not have to be satisfied if $\continuum>\omega_1$.
However, if we consider only $\sigma$-ideal $\I=\I_\omega=[X]^{\leq\omega}$
then the situation simplifies and essentially the original
proof of Theorem \ref{main}(A)
works with the assumption $2^{2^\omega}=(\continuum)^+$ in place of GCH.
The only change in the proof of Theorem \ref{main}(A) that has to be made 
is to define
$\J$ as $[X]^{<2^\omega}$ and remove sets $I_\eta$ from condition (iii).
Thus, we can state this in form of next theorem.

\theorem{mainA}{ Let $|X|=|Y|=\continuum$ and $\R\in[Y^X]^{\leq 2^\omega}$.
If $2^{2^\omega}=(\continuum)^+$
then there is a Hausdorff, connected and
locally connected topology $\tau_2$ on $Y$
such that for every family 
$\F\subset\const\cup\R$ with the property that $\const\subset\F$ and
\[
\{x\in X\colon f(x)=g(x)\}\in[X]^{\leq\omega}
\ \mbox{ for every distinct }\ f,g\in\F 
\]
we have
\[
\F=\C(\tau,\tau_2), 
\]
where $\tau$ is generated by the family
$\{f^{-1}(U)\colon U\in\tau_2, f\in\F\}$. 
Topology $\tau$ is connected and locally connected. It is also Hausdorff, 
provided \F\ separates points.
}

In particular, in Corollaries 
\ref{cor:mainB}, \ref{cor:mainC} and \ref{cor:mainD}
the assumption of GCH can be replaced by the
assumption that $2^{2^\omega}=(\continuum)^+$.

On the other hand, the assumption $2^{2^\omega}=(\continuum)^+$
is fundamental for the proofs of Theorems \ref{main}
and \ref{mainA}.
However, we still are able to get
the following version of Theorem \ref{mainA} without any
extra set theoretical assumption.

\theorem{mainZFC}{  Let $|X|=|Y|=\continuum$ and  $\R\in[Y^X]^{\leq 2^\omega}$.
If $\K\in[Y^X]^{\leq(2^\omega)^+}$
then there is a Hausdorff, connected and
locally connected topology $\tau_2$ on $Y$
such that for every family 
$\F\subset\const\cup\R$ with the property that $\const\subset\F$ and
\[
\{x\in X\colon f(x)=g(x)\}\in[X]^{\leq\omega}
\ \mbox{ for every distinct }\ f,g\in\F 
\]
we have
\begin{equation}\label{conK}
\K\cap\F=\K\cap\C(\tau,\tau_2), 
\end{equation}
where $\tau$ is generated by the family
$\{f^{-1}(U)\colon U\in\tau_2, f\in\F\}$. 
Topology $\tau$ is connected and locally connected. It is also Hausdorff, 
provided \F\ separates points.
}

Proof. We used the assumption $2^{2^\omega}=(\continuum)^+$
in Theorem \ref{mainA} only in (\ref{condGCH}).
However there were two reasons for it's use. The main reason was to 
enumerate all functions from $X^Y$ in a sequence of length of 
$(\continuum)^+$ and make sure that none of the function
from the list is in $\C(\tau,\tau_2)$, provided it is not
in \F. If we list that way all the functions from \K\
than we can deduce (\ref{conK}). 

The second use of $2^{2^\omega}=(\continuum)^+$
in (\ref{condGCH}) was to enumerate $[Y]^{<2^\omega}$
in sequence of length $(\continuum)^+$.
However, we enumerated the sets from 
$[Y]^{<2^\omega}$ to make sure
that they are closed in $(Y,\tau_2)$. But,
the only place in the proof we really needed this fact was
the proof of Lemma \ref{lem:aaa}. Moreover,
we did not need this fact for all sets from $[Y]^{<2^\omega}$
but only for the sets of the particular form of
$h(\{z\in[\e]_X\setminus D\colon 
h(z)\neq f(z) \mbox{ for all } f\in\F_0\})$,
where $D\in[Y]^{\leq\omega}$, $\F_0\in[\F]^{\leq\omega}$ and $h\in X^Y$.
In case of this theorem need to consider only $h$ from \K. Thus,
there is only $(\continuum)^+$-many sets of this form, and we can list all 
of them in a sequence of length $(\continuum)^+$.
This finishes the proof of Theorem \ref{mainZFC}.

\bigskip

Let us notice, that using Theorem \ref{mainZFC}
we can deduce Corollary \ref{cor:Lee} 
without any additional set-theoretical assumptions.

\medskip

As a last position in this section we like to discuss 
separation axioms for topologizing topologies.

A disadvantage of the original form of Theorem \ref{main} is that
for families $\F$ that do not separate points the topology $\tau$ of the
domain is not Hausdorff. Can we modify the theorem to
make topologizing topologies Hausdorff even if $\F$ does no separates
points? The positive answer is given by the next theorem.

\theorem{mainModified}{ Let $|X|=|Y|=\continuum$, $\R\in[Y^X]^{\leq 2^\omega}$
and let \I\
be a proper $\sigma$-ideal on $X$ containing all singletons.

{\rm{\bf (A)}}  If GCH holds
then there are topologies $\tau_1$ and $\tau_2$ on $X$ and $Y$ respectively
such that for every family 
$\F\subset\const\cup\R$ with the property that $\const\subset\F$ and
\[
\{x\in X\colon f(x)=g(x)\}\in\I
\ \mbox{ for every distinct }\ f,g\in\F 
\]
we have
\[
\F=\C(\tau,\tau_2), 
\]
where $\tau$ is generated by the family
$\tau_1\cup\{f^{-1}(U)\colon U\in\tau_2, f\in\F\}$. 
Topologies $\tau_1$, $\tau$ and $\tau_2$ are Hausdorff, 
connected and locally connected.

{\rm{\bf (B)}}  Moreover, it is consistent with the usual axioms of set theory ZFC 
and GCH 
that the topologies $\tau_1$, $\tau$ and $\tau_2$ are completely regular
and Baire.
}

Sketch of the proof. Modify the proofs of either of the part of the
Theorem \ref{main} as follows. Partition $(\continuum)^+$ into 
two sets $C$ and $D$ of the size $(\continuum)^+$ and choose 
$k\colon X\to Y$ such that $|\{x\in X\colon f(x)=k(x)\}|<\continuum$
for every $f\in\R$. In case of proof of part (A) make sure that 
every tuple listed in (\ref{condGCH}) appears for $\eta$ in $C$ and in $D$.
Next, consider $\tau_2$ as topology generated by restricted embedding
$e^\prime\colon Y\to S^C$ of $e$ and $\tau_1$ by restricted
embedding $e_1^\prime\colon X\to S^{\{k\}\times D}$ of $e_1$.

It is not difficult to check that these topologies will have 
the desired properties.

\bigskip

Topologies $\tau$, $\tau_1$ and $\tau_2$ 
Theorem \ref{mainModified} are
Hausdorff and connected. The similar is true for the topologies
of the Theorem \ref{main}, if \F\ separates points. 
Notice also that
from Theorem \ref{th:properties}(v) it is clear that the topologies must be 
connected. 
But, do they have to be Hausdorff? The next theorem gives the negative 
answer to this question.

\theorem{main2}{ Let $|X|=|Y|=\continuum$, $\R\in[Y^X]^{\leq 2^\omega}$and let \I\
be a proper $\sigma$-ideal on $X$ containing all singletons.
If GCH holds
then there is a topology $\tau_2$ on $Y$
such that for every family 
$\F\subset\const\cup\R$ with the property that $\const\subset\F$ and
\[
\{x\in X\colon f(x)=g(x)\}\in\I
\ \mbox{ for every distinct }\ f,g\in\F 
\]
we have
\[
\F=\C(\tau,\tau_2), 
\]
where $\tau$ is generated by the family
$\{f^{-1}(U)\colon U\in\tau_2, f\in\F\}$. 
Topologies $\tau$ and $\tau_2$ are 
connected, locally connected and $T_1$. However, they are not Hausdorff.
}

Proof. Change topology on $P=\{0,1,2\}$ to $\{\emptyset,P,\{0\}\}$.
It is easy to see that it works.

\section{$\!\!\!\!\!\!\!${\bf.} Proof of Theorem 4(B). }\label{SecB}

In this section we will write $H_\omega(A,B)$ for
$\{s\in B^D\colon D\in[A]^{\leq\omega}\}$.

The idea of the proof of Theorem \ref{main}(B)
is essentially the same as that of Theorem \ref{main}(A), 
except that we will take as space $S$ the unit interval $[0,1]$ with
the natural topology and 
we will construct an embedding 
$e\colon Y\to[0,1]^{(2^\omega)^+}$ using the forcing method. 

So, let $V$ be a model of $ZFC$ in which 
$GCH$ holds and let
\[
P=H_\omega(\omega_2,[0,1])
\]
be a forcing notion in $V$ ordered by the reverse inclusion.
Let $G$ be a $V$-generic filter over 
$P$ and put $g=\bigcup G\colon\omega_2\to[0,1]$.
We will show that the statement from the theorem is true in $V[G]=V[g]$.

It is clear that $P$ is $\omega$-closed. In particular, 
the real numbers in $V$ and in $V[g]$ are the
same so, we do not have to be worry about the different
sets of real numbers \reals\ in $V$ and in $V[g]$.
It is also well known (see e.g. \cite{Kunen}) that
under CH forcing $P$ satisfies $\omega_2$-chain condition. So, 
the cardinals are preserved by $P$ and, since
$|H_\omega(\omega_2,[0,1])|=\omega_2$, GCH holds in $V[G]$. 

Take
$X,Y,\I,\R\in V[g]$ as in Theorem \ref{main}.
Clearly it is enough to prove Theorem \ref{main}
for any sets $X$ and $Y$ of cardinality \continuum. In particular,
we can assume that $X,Y\in V$ are the sets of ordinal numbers.
As in the proof of Theorem \ref{main}(A) 
assume that $\R\cap\const=\emptyset$ and
\[
f^{-1}(y)\in\I \mbox{ for all } f\in\R \mbox{ and } y\in Y. 
\]
Define
$\I_0$ as a family of all sets 
$I\in\I$ such that either $|I|=1$ or
$I=\{x\in X\colon f(x)=g(x)\}$ for some $f,g\in\R$, let 
$\J_0$ be a family of all countable unions of sets from $\I_0$ and let
$\J\subset\I$ be the 
$\sigma$-ideal generated by $\J_0$.

Next, choose $\R^\prime\subset Y^X\setminus(\R\cup\const)$ 
of cardinality continuum such that
$|\{x\in X\colon f(x)=g(x)\}|<\continuum$ for every $f\in\R\cup\const$ and 
$g\in\R^\prime$ and such that
\begin{equation}\label{conIzero}
\I_0=\{\{x\in X\colon f(x)=g(x)\}\colon f,g\in\R^\prime,\ f\neq g\}.
\end{equation}
Such a family can be easily chosen by transfinite induction. 
Put $\R_0=\R\cup\R^\prime$.
Notice that if $\F\subset\R^\prime$ 
satisfies the assumption (\ref{conThmain}) of 
Theorem \ref{main}, then so does $\F\cup\R^\prime$.

Since $P$ satisfies $\omega_2$-chain condition, $X,Y\in V$ and
all sets $X,Y,\R_0,\J_0$ have cardinality $\leq\omega_1$
there is $\xi<\omega_2$ such that
$\R_0\cup\{\R_0\}\cup\J_0\cup\{\J_0\}\subset V[g|_\xi]$.
Thus, we can work in extension $V[g|_\xi]$ of $V$ instead in $V$.
To cut unnecessary notational problems we simple 
treat $V[g|_\xi]$ as $V$, i.e, we are assuming that
$\R_0\cup\{\R_0\}\cup\J_0\cup\{\J_0\}\subset V$.

Since $P$ defined as above is isomorphic to
$H_\omega(Y\times\omega_2,[0,1])$
ordered by the reverse inclusion, we also assume that
\[
P=H_\omega(Y\times\omega_2,[0,1]).
\]
Now, if $G$ is a $V$-generic filter over 
$P$ then $g=\bigcup G\colon Y\times\omega_2\to[0,1]$.
We define embedding $e\colon Y\to[0,1]^{\omega_2}$
by $e(y)(\xi)=g(y,\xi)$. 

Let $\F\subset\R_0$, $\F\in V[g]$, be as in Theorem \ref{main}.
Topologies $\tau_2$
and $\tau$ are defined as before. So, it is enough to show that
they satisfy the desired properties.

Since for every $A\in[Y]^{\leq\omega}$ 
and $y\in Y\setminus A$ the set
\begin{eqnarray*}
&
\{s\in P\colon (\exists \alpha<\omega_2)
(s\forces ``A\subset[\{<\alpha,\{0\}>\}]_Y\
 \&\ y\in[\{<\alpha,\{1\}>\}]_Y\mbox{''}\}\\
& = 
\{s\in P\colon(\exists \alpha<\omega_2)(\forall a\in A)
(<a,\alpha,0>,<y,\alpha,1>\in s)\}
\end{eqnarray*}
is dense in $P$, we conclude easily that
$\tau_2$ is Hausdorff and every $A\in[Y]^{\leq\omega}$
is closed in $(Y,\tau_2)$.
Then, it is also obvious that the weak topology $\tau$ on $X$ 
generated by $\F$ 
is Hausdorff if and only if \F\ separates points. 
Also, since $e[Y]$ and $e_1[X]$ are subspaces of
a product of $[0,1]$
we can easily conclude that $(Y,\tau_2)$ and $(X,\tau)$  
are completely regular. 
 
To prove connectedness of these topologies we need some extra facts
and notations. Let $\B_0$ be a countable base for $[0,1]$ and let
$\B_Z$ be the standard base for $[0,1]^Z$ associated with $\B_0$,
i.e., $\B_Z$ is 
the family of all sets
\[
[\delta]=\{g\in[0,1]^Z\colon
(\forall z\in{\rm dom}(\delta))(g(z)\in\delta(z))\},
\] 
where $\delta\in H(Z,\B_0)$.
We will need the following analog of Lemma \ref{lem:connected}
that holds for every $U\in\B_Z$.
\begin{equation}\label{lem2anal}
\mbox{If } S\subset U \mbox{ is disconnected then }
[\varepsilon]\subset U\setminus S \mbox{ for some } 
\varepsilon\in H_\omega(Z,[0,1]), 
\end{equation}
where $[\e]=\{f\in [0,1]^Z\colon \e\subset f\}$.
This is a well known fact and it can be found in \cite{AMillerBaire}
or \cite{KC:Linear}.

Thus, to show that $(Y,\tau_2)$ is connected and locally connected it
is enough to show that $Y\cap[\varepsilon]\neq\emptyset$, 
for every $\varepsilon\in H_\omega(\omega_2,[0,1])$. 
This easily follows from the density, in $P$, of a set
\begin{eqnarray*}
E_\varepsilon & = & \{s\in P\colon (\exists y\in Y)
                 (s\forces ``y\in[\e]\mbox{''})\}\\
& = &
\{s\in P\colon(\exists y\in Y)(\forall \xi\in{\rm dom}(\e))
(<y,\xi,\e(\xi)>\in s)\}
\end{eqnarray*}
for every $\e\in H_\omega(\omega_2,[0,1])$. 

The connectedness and local connectedness of 
$(X,\tau)$ we can be deduced similarly. However we need 
a stronger fact, i.e., condition (\ref{ab})
of Lemma \ref{lem:aaa}. 
We will show that for every $I\in\J$ and 
$\varepsilon\in 
H(\F\times\omega_2,\B_0)$
\begin{equation}\label{conUI}
X\cap[\e]\setminus I
\mbox{ is nonempty and connected in } (X,\tau),
\end{equation}
i.e., that condition (\ref{ab})
of Lemma \ref{lem:aaa} is satisfied for topologies $\tau$ and $\tau_2$.
Notice that it obviously implies that $(X,\tau)$ 
is connected and locally connected.

By (\ref{lem2anal}) in order to prove (\ref{conUI}) it is enough to 
show that
\begin{equation}\label{conUIXX}
X\cap[\varepsilon]\setminus I\neq\emptyset 
\mbox{ for every } I\in\J_0 \mbox{ and }
\e\in H_\omega((\F\cup\R^\prime)\times\omega_2,\B_0).
\end{equation}

Condition (\ref{conUIXX}) follows in natural way from the density
of the sets
\begin{eqnarray*}
E_I^{\varepsilon} & = &\{s\in P\colon (\exists x\in X\setminus I)
                 (s\forces ``x\in[\varepsilon]\mbox{''}\}\\
& = &
\{s\in P\colon(\exists x\in X\setminus I)
(\forall <f,\xi>\in{\rm dom}(\e))
(<f(x),\xi,\varepsilon(f,\xi)>\in s)\}
\end{eqnarray*}
for all $I\in\J_0$ and 
$\e\in H_\omega(\F\times\omega_2,\B_0)$. The density of
this set can be, in turn, deduced from the fact that for every $s\in P$
\[
I^\prime=\{x\in X\colon f(x)=g(x) \mbox{ for } 
<f,\xi>,<g,\xi>\in{\rm dom}(\varepsilon), f\neq g\}\in\J_0,
\]
\[
I^{\prime\prime}=
\{x\in X\colon <f,\xi>\in{\rm dom}(\e)\ \&\ 
<f(x),\xi>\in{\rm dom}(s)\}\in\J_0
\]
so that there exists $x\in X\setminus(I\cup I^\prime\cup I^{\prime\prime})$.

To finish the proof it is enough to show that $\tau$ and $\tau_2$ are Baire and
that $\C(\tau,\tau_2)\subset\F$, since the inclusion
$\F\subset\C(\tau,\tau_2)$ is obvious. 

By way of contradiction let us assume that we can find 
$h\in\C(\tau,\tau_2)\setminus\F$. 
Let $\tau^\prime$ be a weak topology on $X$ generated by $\F\cup\R^\prime$.
Then, in particular, 
$h\in\C(\tau^\prime,\tau_2)\setminus\F$.
We will use  Lemma \ref{lem:aaa}(B) for $h$, topologies $\tau^\prime$
and $\tau_2$ and the $\sigma$-ideal $\J$.
We already checked that $(X,\tau^\prime)$ and $(Y,\tau_2)$
are regular, connected, locally connected, that $Y$ is Hausdorff
and that countable subsets of $Y$ are closed. 
Condition (\ref{ab}) of Lemma \ref{lem:aaa}
is satisfied by (\ref{conUI}), and $\J$ is equal to $\J_0$ from 
the Lemma \ref{lem:aaa} by (\ref{conIzero}).
To use Lemma \ref{lem:aaa}(B) we have to show that
$\bigcap_{n<\omega}[\varepsilon_n]\neq\emptyset$
for every sequence 
$\{\varepsilon_n\in H((\F\cup\R^\prime)\times\omega_2,\B_0)\colon n<\omega\}$ 
such that ${\rm cl}_X([\varepsilon_{n+1}])\subset[\varepsilon_n]$.
But it is easy to see that sets ${\rm cl}([\varepsilon_n])$,
considered as subsets of the entire 
$[0,1]^Z$, are compact, so they have nonempty intersection. However, 
we used only countable many coordinates in definitions of
$\bigcap_{n<\omega}{\rm cl}([\varepsilon_n])$. So, there exists
$\delta\in H_\omega((\F\cup\R^\prime)\times\omega_2,[0,1])$ such that
$[\delta]\subset\bigcap_{n<\omega}{\rm cl}([\varepsilon_n])=
\bigcap_{n<\omega}[\varepsilon_n]$. This, and (\ref{conUIXX}),
imply (B) of Lemma~\ref{lem:aaa}. This also imply easily that $(X,\tau)$ 
is Baire. The proof that $(Y,\tau_2)$ is Baire is similar. 

So, let $x_0$ be as in Lemma \ref{lem:aaa}(B) for $h$. 
Similarly as in the proof of Theorem~\ref{main}(A) we will show that 
there exists an ordinal $\eta<\omega_2$ such that 
$x_0\in h^{-1}([\{<\eta,\{0\}>\}])$ and that
$[\varepsilon]\cap h^{-1}([\{<\eta,\{1\}>\}])\neq\emptyset$
for every $\varepsilon\in H(\F\times\omega_2,\B_0)$ with $x_0\in[\varepsilon]$.
This will give as a contradiction with the fact that 
$h\in\C(\tau,\tau_2)$, since \mbox{$h^{-1}([\{<\eta,[0,1/2)>\}])$}
would be nonempty open set with empty interior.

By $\omega_2$-chain condition of forcing $P$ we can find $\zeta<\omega_2$
such that $h\in V[g|_{Y\times\zeta}]$. Since now we will work
in $V[g|_{Y\times\zeta}]$ with forcing notion 
$P_1=H_\omega(Y\times(\omega_2\setminus\zeta),[0,1])$.
Clearly the set
\begin{eqnarray*}
E_0 & = &\{s\in P_1\colon (\exists \eta\in\omega_2\setminus\zeta)
                 (s\forces ``h(x_0)\in[\{<\eta,\{0\}>\}]\mbox{''})\}\\
& = &
\{s\in P_1\colon(\exists \eta\in\omega_2\setminus\zeta)(<h(x_0),\eta,0>\in s)\}
\end{eqnarray*}
is dense in $P_1$. So, there exists $\eta\in\omega_2\setminus\zeta$
such that $x_0\in h^{-1}([\{<\eta,\{0\}>\}])$. 
Let $s_0\in P_1$ be such that 
$s_0\forces ``x_0\in h^{-1}([\{<\eta,\{0\}>\}])\mbox{''}$.
To show that
\[
[\varepsilon]\cap h^{-1}([\{<\eta,\{1\}>\}])\neq\emptyset
\]
for every $\varepsilon\in H(\F\times\omega_2,\B_0)$ with $x_0\in[\varepsilon]$
fix such an $\varepsilon$. To finish the proof it is enough to show
that the set
\begin{eqnarray*}
E & = & \{s\in P_1\colon (\exists x\in X) 
    (s\forces ``x\in[\varepsilon]\cap h^{-1}([\{<\eta,\{1\}>\}])\mbox{''})\}\\
  & = &
   \{s\in P_1\colon  (\exists x\in[\varepsilon|_{{\cal F}\times\zeta}])
       ((<h(x),\eta,1>\in s)\ \&   \\
  &   & (\forall <f,\xi>\in{\rm dom}(\varepsilon), \xi\geq\zeta)
(s(f(x),\xi)\in\varepsilon(f(x),\xi))\}.
\end{eqnarray*}
is dense in $P_1$ below $s_0$. 

To see it, choose $t\in P_1$, $t\leq s_0$. 
We must find $s\leq t$, $s\in P_1$ 
and an $x\in[\e|_{{\cal F}\times\zeta}]$
such that $<h(x),\eta>\in{\rm dom}(s)$, $s(h(x),\eta)=1$ and
for every $<f,\xi>\in{\rm dom}(\varepsilon)$, $\xi\geq\zeta$,
we have $<f(x),\xi>\in{\rm dom}(s)$
with $s(f(x),\xi)\in\e(f(x),\xi)$.
But let $\F_0$ be the set of all $f$ such that either
$<f,\alpha>\in{\rm dom}(\e)$ for some $\alpha$ or $f$ is equal to
a constant $m\in M$,
where $M=\{c\colon <c,\beta>\in{\rm dom}(t)\mbox{ for some }\beta\}$.
Then, 
it is enough to find $x\in[\e|_{{\cal F}\times\zeta}]$
such that $x$ does not belong to
\[
I=\{z\in X\colon(\exists f,g\in\F_0)(f(z)=g(z)\ \&\ f\neq g)\}\in\J,
\]
(i.e., that $f(x)\neq g(x)$ and $f(x)\not\in M$ for all
$<f,\xi>,<g,\xi>\in{\rm dom}(\e)$, $f\neq g$, $\xi\geq\zeta$),
and that 
\[
h(x)\in 
h(\{z\in[\e|_{{\cal F}\times\zeta}]\setminus I\colon h(z)\neq f(z) 
                  \mbox{ for all } f\in\F_0\})\setminus M.
\]
But this can be done by the conclusion of Lemma \ref{lem:aaa}(B).

Theorem \ref{main}(B) has been proved.

\bigskip

We will finish this paper with the following two problems.


\problem{pr:ZFC}{ Can we prove Theorem \ref{main} or any of the Corollaries
\ref{cor:mainA}, \ref{cor:mainB}, \ref{cor:mainC}, \ref{cor:mainD}
without any additional set-theoretical assumptions?}

\problem{pr:reg}{ Can topologies from 
Theorem \ref{main} or any of the Corollaries
\ref{cor:mainA}, \ref{cor:mainB}, \ref{cor:mainC}, \ref{cor:mainD}
be normal? Lindel\"of? hereditarily Lindel\"of? compact? metrizable?}



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