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\put(143,66){\makebox(0,0)[tl]{\footnotesize Proceedings of the Ninth Prague Topological Symposium}}
\put(143,50){\makebox(0,0)[tl]{\footnotesize Contributed papers from the symposium held in}}
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\title[Characterizing continuity]{Characterizing continuity by preserving
compactness and connectedness}
\author[J. Gerlits]{J\'anos Gerlits}
\address{Alfr\'ed R\'enyi Institute of Mathematics\\
P.O.Box 127, 1364 Budapest, Hungary}
\email{gerlits@renyi.hu}
\author[I. Juh\'asz]{Istv\'an Juh\'asz}
\address{Alfr\'ed R\'enyi Institute of Mathematics\\
P.O.Box 127, 1364 Budapest, Hungary}
\email{juhasz@renyi.hu}
\author[L. Soukup]{Lajos Soukup}
\address{Alfr\'ed R\'enyi Institute of Mathematics\\
P.O.Box 127, 1364 Budapest, Hungary}
\email{soukup@renyi.hu}
\author[Z. Szentmikl\'ossy]{Zolt\'an Szentmikl\'ossy}
\address{E\"otv\"os Lor\'ant University, Department of Analysis,
1117 Budapest, P\'azm\'any P\'eter s\'et\'any 1/A, Hungary}
\email{zoli@renyi.hu}
\keywords{Hausdorff space, continuity, compact, connected, locally
connected, Fr\`echet space, monotonically normal, linearly ordered space}
\subjclass[2000]{54C05, 54D05, 54F05, 54B10}
\thanks{The second author was an invited speaker at the Ninth Prague Topological Symposium.}
\thanks{Research supported by Hungarian Foundation for Scientific
Research, grant No.25745.}
\thanks{This article has been submitted for publication to
\textit{Fundamenta Mathematicae}.}
\thanks{J\'anos Gerlits, Istv\'an Juh\'asz, Lajos Soukup and Zolt\'an
Szentmikl\'ossy,
{\em Characterizing continuity by preserving compactness and
connectedness},
Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001),
pp.~93--118, Topology Atlas, Toronto, 2002; {\tt arXiv:math.GN/0204125}}
\begin{abstract}
Let us call a function $f$ from a space $X$ into a space
$Y$ {\it preserving} if the image of every compact subspace of $X$
is compact in $Y$ and the image of every connected subspace of $X$
is connected in $Y$. By elementary theorems a continuous function
is always preserving. Evelyn R. McMillan \cite{McM} proved in 1970
that if $X$ is Hausdorff, locally connected and Fr\`echet, $Y$ is
Hausdorff, then the converse is also true: any preserving function
$f:X\to Y$ is continuous.
The main result of this paper is that if $X$ is any product of
connected linearly ordered spaces (e.g.\ if $X={\mathbb R}^\kappa$) and
$f:X \to Y$ is a preserving function into a
regular space $Y$, then $f$ is continuous.
\end{abstract}
\maketitle
Let us call a function $f$ from a space $X$ into a space $Y$ {\it
preserving} if the image of every compact subspace of $X$ is
compact in $Y$ and the image of every connected subspace of $X$ is
connected in $Y$. By elementary theorems a continuous function is
always preserving. Quite a few authors noticed---mostly
independently from each other---that the converse is also true for
real functions: a preserving function $f:{\mathbb R} \to
{\mathbb R}$ is continuous. (The first paper we know of is
\cite {Rowe}
from 1926!) Whyburn proved \cite{Wh} that a preserving function
from a space $X$
into a Hausdorff space is always continuous at a first countability and
local connectivity point of $X$.
Evelyn R. McMillan \cite{McM} proved in
1970 that if $X$ is Hausdorff, locally connected and Fr\`echet,
moreover $Y$ is Hausdorff, then any preserving function $f:X\to Y$ is
continuous. This is quite a significant and deep result that is
surprisingly little known.
We shall use the notation $Pr(X,T_i)$
($i=1,2,3$ or $3{\frac{1}{2}}$) to denote the following statement:
Every preserving function from the topological space $X$ into
any $T_i$ space is continuous.
The organization of the paper is as follows: In \S $1$ we give some basic
definitions and then treat our results that are closely connected to
McMillan's theorem.
\S $2$ treats several important technical theorems that enable us to
conclude that certain preserving functions are continuous.
In \S $3$ we prove that certain product spaces $X$ satisfy $Pr(X,T_3)$; in
particular, any preserving function from a product of connected linearly
ordered spaces into a regular space is continuous.
In \S $4$ we discuss some results concerning the continuity of preserving
functions defined on compact or sequential spaces.
Finally, \S $5$ treats the relation $Pr(X,T_1)$.
Our terminology is standard.
Undefined terms can be found in \cite{Eng} or in \cite{Juh}.
\section{Around McMillan's theorem}
Our first theorem implies that (at least among Tychonoff spaces) local
connectivity of $X$ is a necessary condition for
$Pr(X,T_{3{\frac{1}{2}}})$.
For metric spaces this result was proved by Klee and Utz\cite{Kl}.
\begin{theorem}
\label{1.1}
If the Tychonoff space $X$ is not locally connected at a point $p \in X$,
then there exists a preserving function $f$ from $X$ into the interval
$[0,1]$ which is not continuous at $p$.
\end{theorem}
\begin{proof}
Suppose $X$ is not locally connected at the point $p$, then there is an
open neighbourhood $U$ of $p$ such that if $K$ denotes the component of
$p$ in $U$, then $K$ is not a neighbourhood of $p$. As $X$ is Tychonoff,
there exists an open neighbourhood $V\subset U$ of $p$ and a continuous
function $\overline f:X \to [0,1]$ such that $\overline f$ is identically
$0$ on $V$ and identically $1$ outside of $U$. Select another continuous
function $g:X \to [0,1]$ such that $g(p)=1$ and $g$ is identically $0$
outside $V$. Put now
\begin{displaymath}
f(x)=
\left\{
\begin{array}{ll}
\overline{f}(x) + g(x)&\text{ if $x\in K$;}\\
\overline {f}(x) & \text{otherwise.}
\end{array}
\right.
\end{displaymath}
($f:X \to [0,1]$ because $\overline f[V]=\{0\}$ and $g[X-V]=\{0\}$.)
The function $f$ is not continuous at $p$ because $f(p)=1$ and in every
neighbourhood of $p$ there is a point from $V - K$ mapped to $0$.
On the other hand, we claim that $f$ is preserving. Indeed, let $C \subset
X$ be compact. The restriction of $f$ to the closed set $F=(X-V)\cup K$ is
evidently continuous (here $f=\overline f + g$), hence $C'=f(C\cap F)$ is
compact. But $f$ is $0$ on $V-K$ and so $f(C)$ is either $C'$ or $C' \cup
\{0\}$, thus $f(C)$ is clearly compact.
Let now $C$ be any connected subset of $X$. The function $f$ is continuous
on $X-K$ (here $f=\overline f$), hence $C\cap K \not = \emptyset$ can be
assumed. Similarly, $f$ is continuous on $X-V$ ($f(x)=\overline f(x)$ for
$x\in X-V$), hence we can suppose that $C$ also meets $V$. If $C \subset
U$, then necessarily $C \subset K$ because $C$ is connected and meets the
component $K$of $U$. As $f$ is continuous on $K$, the image $f(C)$ is then
connected.
Thus it remains to check only the case in which $C$ meets both $V$ and
$X-U$. In this case $\overline f(C)$ is the whole interval $[0,1]$. But
$f$ and $\overline f$ are equal at each point where $\overline f$ does not
vanish, thus $f(C)$ contains the interval $(0,1]$ and so it is connected.
\end{proof}
It is not a coincidence that the target space in Theorem \ref{1.1} is
the interval $[0,1]$, because of the following result:
\begin{lemma}
\label{1.2}
Suppose $f: X \to Y$ is a preserving function into a Tychonoff space $Y$
and $f$ is not continuous at the point $p \in X$. Then there exists a
preserving function $h:X \to [0,1]$ which is also not continuous at $p$.
\end{lemma}
\begin{proof}
Since $f$ is not continuous at $p$, there exists a closed set $F\subset Y$
such that $f(p)\not \in F$ but $p$ is an accumulation point of
$f^{-1}(F)$.
Choose a continuous function $g:Y\to [0,1]$ such that $g(f(p))=0$ and $g$
is identically $1$ on $F$. Then the composite function $h(x)=f(g(x))$ has
the stated properties.
\end{proof}
The following Lemmas will be often used in the sequel.
They all state simple properties of preserving functions.
\begin{lemma}
\label{1.3}If $f:X \to Y$ is a
compactness preserving function, $Y$ is Hausdorff, $M
\subset X$ with $\overline M$ compact and $f(M)$
infinite then for every accumulation point $y$ of $f(M)$
there is an accumulation point $x$ of $M$ such that
$f(x) = y$, i. e. $f(M)' \subset f(M')$ .
\end{lemma}
\begin{proof}
Let $N = M - f^{-1}(y)$ then $f(N) = f(M) - \{y\}$ and so we have
$y\in \overline {f(N)}- f(N)$. But $f(\overline {N})$ is also compact,
hence closed in $Y$ and so $y\in f(\overline {N}) - f(N)$ as well.
Thus there is an $x\in \overline N - N$ such that $f(x)=y$ and then $x$ is
as required.
\end{proof}
We shall often use the following immediate consequence of this lemma:
\begin{qlemma}[E.~R.~McMillan \cite{McM}]
\label{1.3'}
If $f:X \to Y$ is a compactness preserving function, $Y$ is Hausdorff,
$\{x_n : n<\omega \} \subset X$ converges to $x\in X$ then either
${\{ f(x_n): n<\omega \}}$, converges to $f(x)$ or the set
$\{f(x_n): n<\omega \}$ is finite
\easytosee
\end{qlemma}
Actually, to prove Lemma \ref{1.3'} we do not need the full force of the
assumption that $f$ is compactness preserving. It suffices to assume that
the image of a convergent sequence together with its limit is compact, in
other words: the image of a topological copy of $\omega +1$ is compact.
For almost all of our results given below only this very restricted
special case of compactness preservation is needed.
\begin{lemma}[\cite{Per}]
\label{1.4}
If $f:X \to Y$ preserves connectedness, $Y$ is a $T_1$-space and $C\subset
X$ is a connected set, then $f(\overline C) \subset \overline {f(C)}$.
\end{lemma}
\begin{proof}
If $x\in \overline C$ then $C\cup \{x\}$ is connected.
Thus $f(C\cup \{x\})= f(C)\cup \{f(x)\}$ is also connected and hence
$f(x) \in \overline {f(C)}$.
\end{proof}
The next lemma will also play a crucial role in some theorems of the
paper. A weaker form of it appears in \cite{McM}.
\begin{definition}
\label{1.5}
We shall say that $f:X \to Y$ is {\it locally constant} at the point
$x \in X$ if there is a neighbourhood $U$ of $x$ such that $f$ is constant
on $U$.
\end{definition}
\begin{lemma}
\label{1.6}
Let $f$ be a connectivity preserving function from a locally connected
space $X$ into a $T_1$-space $Y$. If $F\subset Y$ is closed and $p \in
\overline {f^{-1}(F)} - f^{-1}(F)$ then $p$ is also in the closure of
the set
\begin{displaymath}
\{ x \in f^{-1}(F): f \hbox { is not locally constant at
} x \}.
\end{displaymath}
\end{lemma}
\begin{proof}
Let $G$ be a connected open neighbourhood of $p$ and $C$ be
a component of the non-empty subspace $G \cap f^{-1}(F)$. Then $C$ has a
boundary point $x$ in the connected subspace $G$ because $\emptyset \not =
C \not = G$. Clearly, $f(x) \in F$ by Lemma \ref{1.4}. If $V\subset G$ is
any connected neighbourhood of $x$ then $V \cup C$ is connected and $V-C
\not = \emptyset$ because $x$ is a boundary point of $C$ hence $V$ is not
contained in $f^{-1}(F)$, so $f$ is not locally constant at $x$.
\end{proof}
\begin{lemma}
\label{1.7}
Let $f:X \to Y$ be a connectivity preserving function into the $T_1$-space
$Y$.
Suppose that $X$ is locally connected at the point $p \in X$ and $f$ is
not locally constant at $p$.
Then $f(U)\cap V$ is infinite for every neighbourhood $U$ of $p$ and for
every neighbourhood $V$ of $f(p)$.
\end{lemma}
\begin{proof}
Choose any connected neighbourhood $U$ of $x$; then $f(U)$ is connected
and has at least two points. Thus if $V$ is any open subset of $Y$
containing $f(p)$ then $f(U) \cap V$ can not be finite because otherwise
$f(p)$ would be an isolated point of the non-singleton connected set
$f(U)$.
\end{proof}
The following proof of McMillan's theorem is based upon the same ideas as
her original proof, although, we think, it is much simpler.
We include it here mainly to make the paper self-contained.
\begin{theorem}[E.~R.~McMillan \cite{McM}]
\label{1.8}
If $X$ is a locally connected and Fr\`echet Hausdorff space,
then $Pr(X,T_2)$ holds.
\end{theorem}
\begin{proof} Assume $Y$ is $T_2$ and $f: X \to Y$ is
preserving but not continuous at the point $p \in X$. Then by
Lemma \ref{1.3'} there is a sequence $x_n \to p$ such that $f(x_n) = y
\not = f(p)$ for all $n<\omega$ . Using Lemma \ref{1.6}
with $F = \{y\}$ we can also
assume that $f$ is not locally constant at the points $x_n$.
As $Y$ is $T_2$, there is an open set $V \subset Y$ such that $y
\in V$ but $f(p) \not \in \overline V$. By Lemma \ref{1.7} the image of
every neighbourhood of each point $x_n$ contains infinitely many
points (different from $y$) from $V$.
Now we select recursively sequences $\{x^n_k : k<\omega \}$
converging to $x_n$ for all $n<\omega$. Suppose $n<\omega$ and
the points $x^m_k$ are already defined for $m\max (n_l,k_l)$ for all
$l<\omega$. However, the sequence $\{f(x^{n_l}_{k_l}) : l<\omega \}$
does not converge to $f(p)$ because
$f(p)\not \in \overline {\{f(x^{n_l}_{k_l}):
l<\omega \}} \subset \overline V $,
while the points $f(x^{n_l}_{k_l})$ are all
distinct, contradicting Lemma \ref{1.3'}. \end{proof}
We could prove the following local version of McMillan's theorem:
\begin{theorem}
\label{1.9}
If $X$ is a locally connected Hausdorff space, $p$ is a Fr\`echet point of
$X$ and $f$ is a preserving function from $X$ into a Tychonoff space $Y$,
then $f$ is continuous at $p$.
\end{theorem}
\begin{proof}
By Lemma \ref{1.2} it suffices to prove this in the case
when $Y$ is the interval $[0,1]$.
Assume, indirectly, that $f$ is not continuous at $p$
then, since $p$ is a Fr\'echet point and by Lemma \ref{1.6}, we can choose a
sequence $x_n \to p$ and a $y \in [0,1]$ with $y\not= f(p)$
such that $f(x_n) = y$ and $f$ is not locally constant at $x_n$
for all $n < \omega$.
For each $n$ choose a neighbourhood $U_n$ of $x_n$ with $p \not \in
\overline U_n$ and put $A_n = \{x\in U_n : 0<|f(x)-y|<1/n \}$. For any
connected neighbourhood $W$ of $x_n$ its image $f(W)$ is a non-singleton
interval containing $y$, hence the local connectivity of $X$ implies that
$x_n \in \overline A_n$ for all $n<\omega$ and so $p$ belongs to the
closure of $\bigcup \{A_n : n<\omega \}$. As $p$ is a Fr\`echet point,
there is a sequence $z_k \in A_{n_k}$ converging to $p$. But $n_k$
necessarily tends to infinity because $p \not \in \overline A_n \subset
\overline U_n$ for each $n < \omega$, hence $f(z_k) \to y \not = f(p)$,
contradicting Lemma \ref{1.3'}. Indeed, the set $\{f(z_k) : k< \omega \}$
is infinite because $f(z_k)\not = y$.
\end{proof}
Theorem \ref{1.9} is not a full local version of Theorem \ref{1.8}
because local connectivity is assumed in it globally for X.
This leads to the following natural question:
\begin{problem}
\label{1.10} Let $X$ be a Hausdorff space,
$f$ be a preserving function from $X$ into a Tychonoff space $Y$
and let $X$ be locally connected and Fr\`echet at the point $p\in
X$. Is it true then that $f$ is continuous at $p$?
\end{problem}
We do not know the answer to this problem, however
we can prove some partial affirmative results.
\begin{definition}[\cite{Arh}]
\label{1.11} A point $x$ of
a space $X$ is called an $(\alpha_4)$ point if for any sequence
$\{A_n :n<\omega \}$ of countably infinite sets with $A_n \to x$
for each $n<\omega$ there is a countably infinite set $B \to x$
such that $\{n<\omega: A_n \cap B \not = \emptyset \}$ is
infinite.
An $(\alpha_4)$ and Fr\`echet point will be called an
$(\alpha_4)$-F point in $X$.
\end{definition}
\begin{theorem}
\label{1.12} Let $f$ be a preserving
function from a topological space $X$ into a Hausdorff space $Y$
and let $p$ be a point of local connectivity and an $(\alpha_4)$-F
point in $X$. Then $f$ is continuous at $p$.
\end{theorem}
\begin{proof} Assume not. Then by the Lemma \ref{1.3'}
there is a point $y\in Y$ such that $y\not = f(p)$ but $p$ is in
the closure of $f^{-1}(y)$. Choose an open neighbourhood $V$
of $y$ in $Y$ with $f(p) \not \in \overline V$. By Lemma \ref{1.7}
and Lemma \ref{1.3'} we can recursively choose
pairwise distinct points
$y_n\in V$ such that $p$ is in the closure of $f^{-1}(y_n)$
for all $n\in \omega$. As the point $p$ is an $(\alpha_4)$-F
point in $X$, there is a ``diagonal'' sequence $\{x_m:m\in M \}$
converging to $p$, where $f(x_m)=y_m$ and $M\subset \omega$ is
infinite, contradicting Lemma \ref{1.3'}.\end{proof}
The next result yields a different kind of
partial answer to Problem \ref{1.10}:
\begin{theorem}
\label{1.13} Let $f$ be a preserving
function from a topological space $X$ into a Tychonoff space $Y$
and let $p$ be a Fr\` echet point of local connectivity of $X$
with character $\le 2^{\omega}$. Then $f$ is continuous at $p$.
\end{theorem}
\begin{proof} Assume not, $f$ is discontinuous at the
point $p \in X$. By Lemmas \ref{1.2} and \ref{1.3'}
we can suppose that $Y=[0,1]$,
$f(p)=0$ and every neighbourhood of $p$ is mapped onto the whole
interval $[0,1]$ .
Let $\mathcal U$ be a neighbourhood-base of $p$ of size $\le
2^{\omega}$ and choose for each $U \in \mathcal U$ a point $x_U \in U$
such that $f(x_U)\in [1/2, 1]$ and the points $f(x_U)$ are all
distinct.
Put $A=\{ x_U : U \in {\mathcal U} \}$, then $p \in \overline A$ and
so there exists a sequence $\{x_n : n<\omega \} \subset A$
converging to $p$, contradicting Lemma \ref{1.3'}.
\end{proof}
There is a variant of this result in which the assumption that
$Y$ be Tychonoff is relaxed to $T_3$, however the assumption on
the character of the point $p$ is more stringent.
Its proof will make
use of the following (probably well-known) lemma:
\begin{lemma}
\label{1.14} Let $Z$ be an infinite connected
regular space, then any non-empty open subset $G$ of $Z$
is uncountable.
\end{lemma}
\begin{proof} Choose a point $z \in G$ and an open
proper subset $V$ of $G$ with $z\in V \subset \overline V \subset G$.
If $G$ would be countable then, as a countable
regular space, $G$ would be Tychonoff, and so there would be a continuous
function $f: G \to [0,1]$ such that $f(z)=1$ and
$f$ is identically zero on $G-V$. Extend $f$ to a function
$\overline f : Z \to [0,1]$ by putting $\overline f(y)=0$ if $y\in
Z-G$. Then $\overline f$ is continuous and hence $\overline {f}(Z)$ is
also connected.
Consequently we have $f(G) = \overline{f}(Z)=[0,1]$ implying
that $|G|\ge |[0,1]|>
\omega$, and so contradicting that $G$ is countable.\end{proof}
\begin{theorem}
\label{1.15} Let $f$ be a preserving
function from a topological space $X$ into a regular space $Y$ and
let $p \in X$ be a Fr\` echet point of local connectivity with
character $\le \omega_1$. Then $f$ is continuous at $p$.
\end{theorem}
\begin{proof} Assume $f$ is discontinuous at the
point $p \in X$. As $p$ is a a Fr\` echet point, there is a
sequence $x_n \to p$ such that $f(x_n)$ does not converge to
$f(p)$. Taking a subsequence if necessary, we can suppose by Lemma
\ref{1.3'} that $f(x_n)=y \ne f(p)$ for all $n<\omega$.
Choose now an open neighbourhood $V$ of the point $y\in Y$ with
$f(p) \not \in \overline V$. Let $\mathcal U$ be a neighbourhood base
of the point $p$ in $X$ such that $|{\mathcal U}|\le \omega_1$ and the
elements of $\mathcal U$ are connected.
Choose now points $x_U$ from the sets $U \in \mathcal U$ such that
$f(x_U)\in V$ and the points $f(x_U)$ are all distinct. This can
be accomplished by an easy transfinite recursion because for each
$U\in \mathcal U$ the set $f(U)$ is connected and infinite, hence
$f(U)\cap V$ is uncountable by the previous
lemma. Put $A=\{x_U : U \in {\mathcal U} \}$. Then $p \in \overline A$
and so there exists a sequence $\{y_n : n<\omega \} \subset A$
converging to $p$, contradicting Lemma \ref{1.3'}.
\end{proof}
We shall now consider some further topological properties and prove
several results about them saying that preserving functions are
sequentially continuous.
Since in a Fr\'echet point sequential continuity implies continuity, these
results are clearly relevant to McMillan's theorem.
Their real significance, however, will only become clear in the following
two sections.
\begin{definition}
\label{1.16} A point $x$ in a
topological space $X$ is called a {\it sequentially connectible
(in short: SC)} point, if $x_n \in X$, $x_n \to x$ implies that there are an
infinite subsequence $\langle x_{n_k} : k<\omega \rangle $ and a
sequence $\langle C_k :k<\omega \rangle $ consisting of connected
subsets of $X$ such that $\{x_{n_k},x\} \subset C_k$ for all $k<\omega$,
(i.e.\ $C_k$ ``connects'' $x_{n_k}$ with $x$, this explains the
terminology,)
moreover $C_k \to x$, i.e.\ every neighbourhood of
the point $x$ contains all but finitely many $C_k$'s.
A space $X$ is called
an $SC$ space if all its points are $SC$ points.
\end{definition}
\begin{remark}
\label{1.17}
It is clear that the SC property is closely related to local connectivity.
Let us say that a point $x$ in
space $X$ is a strong local connectivity point if it
has a neighbourhood base $\mathcal B$ such that the
intersection of an arbitrary (non-empty) subfamily of $\mathcal B$ is
connected. For example, local connectivity points
of countable character
or any point of a connected linearly ordered space has this
property.
We claim that {\it if $x$ is a strong local connectivity point of $X$
then $x$ is an $SC$ point in $X$.} Indeed,
assume that $x_n \to x$ and for every $n
\in \omega$ let $C_n$ denote the intersection of all those members of
$\mathcal B$ which contain both points $x_n$ and $x$.
(As the sequence $\{ x_n : n<\omega \}$ converges to $x$, we can
suppose that some element $B_0 \in {\mathcal B }$ contains all the
$x_n$'s.) Then $\{x,x_n\} \subset C_n$, moreover $C_n \to x$. Indeed,
the latter holds because if
$x\in B \in {\mathcal B}$ then, by definition, $x_n \in B$ implies
$C_n \subset B$. \easytosee
\end{remark}
The $SC$ property does not imply local connectivity. (If every convergent
sequence is eventually constant then the space is trivially $SC$.)
However, the following simple lemma shows that if there are
``many'' convergent sequences then such an implication is valid.
\begin{lemma}
\label{1.18}
Let $x$ be a both Fr\`echet and $SC$ point in a space $X$.
Then $x$ is also a point of local connectivity in $X$.
\end{lemma}
\begin{proof}
Let $G$ be any open set containing $x$ and set $$H = \cup \{\,C
:\, x \in C \subset G\,\,\, \hbox{and}\,\, C\,\, \hbox{is
connected} \,\}.$$ We claim that (the obviously connected) set $H$
is a neighbourhood of $x$. Indeed, otherwise, as $x$ is a
Fr\`echet point, we could choose a sequence $x_n \to x$ from the
set $G-H$ while for every point $y \in G-H$ no connected set
containing both $x$ and $y$ is a subset of $G$, contradicting the
$SC$ property of $x$.
\end{proof}
If $SC$ holds globally, i.e.\ in an $SC$ space, then in the above result
the Fr\`echet property can be replaced with a weaker property that will
turn out to play a very important role in the sequel.
\begin{definition}
\label{1.19}
A point $p$ in a topological space $X$ is called an {\it s} point if for
every family $\mathcal A$ of subsets of $X$ such that
$p \in \overline {\bigcup \mathcal A}$ but $p \not \in \overline A$ for
all $A \in \mathcal A$ there is a sequence
$\langle \langle x_n,A_n \rangle :n<\omega \rangle$ such that
$x_n \in A_n \in \mathcal A$, the sets $A_n$ are pairwise distinct and
$\{x_n \}$ converges to some point $x \in X$ (that may be different from
$p$).
\end{definition}
A Fr\`echet point is evidently an $s$ point, moreover any point
that has a sequentially compact neighbourhood is also an $s$ point.
Other examples of $s$ points will be seen later.
\begin{theorem}
\label{1.20}
Any $s$ point in a $T_3$ and $SC$ space is a point of local connectivity.
\end{theorem}
\begin{proof}
Let $p$ be an $s$ point in the regular $SC$ space $X$ and let $G$ be an
open neighbourhood of $p$.
We have to prove that the component $K_0$ of the point $p$ in $G$ is a
neighbourhood of $p$.
Assume this is false and choose an open set $U$ such that $p \in U \subset
\overline U \subset G$.
Put
\begin{displaymath}
{\mathcal A }= \{K\cap \overline U : K \hbox{ is a component of
}G,\,\, K\not = K_0 \}.
\end{displaymath}
Then $p \in \overline {\bigcup \mathcal A}$ and $p \not \in \overline
A$ for $A\in \mathcal A$ (because a component of $G$ is relatively
closed in $G$), hence, by the definition of an $s$ point, there
exists a sequence $\{ \langle x_n, A_n \rangle : n<\omega \}$
such that $x_n \in A_n \in \mathcal A$, $x_n \to x$ for some $x\in X$
and if $A_n = K_n \cap \overline U$ then the components $K_n$ are
distinct. Note that $x\in \overline U \subset G$. As distinct
components are disjoint, we can assume that $x \not \in K_n$ for all
$n<\omega$. As $x$ is an $SC$ point, there are a connected
set $C$ and some $n<\omega$ such that $\{x, x_n \}\subset C\subset
G$. However, this is impossible, because then $K_n \cup C$ would
be a connected set in $G$ larger then the component
$K_n$.\end{proof}
The significance of the $SC$ property in our study of
continuity properties of preserving functions is revealed by the
following result.
\begin{theorem}
\label{1.21}
A preserving function $f:X \to Y$ into a Hausdorff space $Y$ is
sequentially continuous at each $SC$ point of $X$.
\end{theorem}
\begin{proof}
Let $x \in X$ be an $SC$ point and assume that $x_n \to x$ but $f(x_n)$
does not converge to $f(x)$ for a sequence $\{x_n : n< \omega \}$ in $X$.
We can assume by Lemma \ref{1.3'} that $f(x_n)=y\not = f(x)$ for all
$n<\omega$.
Choose an open neighbourhood $V$ of $y$ in $Y$ such that
$f(x)\not \in \overline V$.
As $x$ is an $SC$ point in $X$, we can also assume that there is a
sequence of connected sets $C_n$ such that $C_n \to x$ and $x,x_n \in C_n$
for $n<\omega$.
We can now define a sequence $z_n \in C_n$ such that $f(z_n) \in V$ and
the points $f(z_n)$ are all distinct.
Indeed, assume $n<\omega$ and the points $z_i$ are already defined for
$i 2^{\omega}$'' of our result is satisfied if $p=
2^{\omega}$ (hence Martin's axiom implies it), but it is also true if
$2^{\omega_1} > 2^{\omega}$.
Now, our promised consistency result on compact sequential spaces
will be a corollary of a ZFC result of somewhat technical nature.
Before formulating this, however, we shall prove two
lemmas that may have some independent interest in themselves.
\begin{lemma}
\label{4.6} Let $X$ be a compact $T_2$
space of countable tightness and $f : X \to [0,1]$ be a compactness
preserving map of $X$ into the unit interval. If $x \in X$ is a point
in $X$ and $[a,b]$ is a subinterval of $[0,1]$ such that for every
neighbourhood $U$ of $x$ we have $[a,b] \subset f(U)$ then for any
$G_{