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\begin{document}
\chapter{\v{C}ech Topologies -- a Survey}
\markboth{\v{C}ech Topologies -- a Survey}{\v{C}ech Topologies -- a Survey}
\markright{\v{C}ech Topologies -- a Survey}
\thispagestyle{empty}
\v{C}ech topologies were introduced into rough set theory by
Marcus [7], who extended the natural correspondence between
similarity relations and \v{C}ech topologics rough set theory.
In this Chapter, we collect results about \v{C}ech topologies,
otherwise exposed in a specialized monograph [3].
\section{Introduction}
Eduard \v{C}ech (see [3]) introduced a new notion of a
topology, weaker than the
one proposed among others, by M.Frech\'{e}t F.Riesz, F.Hausdorf,
K.Kuratowski and more suitable in many contexts, when one wants
to introduce a closure operator but the conditions do not permit
such an operator to be a classical closure operator. The striking
example is concerned with the opposition tolerance - equivalence:
an equivalence relation $R$ introduces the closure
operator $cl_{R}$, expressing the closure of a set $A$ as the
union of equivalence classes of $R$ that intersect $A$. A~to\-le\-ran\-ce
relation $J$, with equivalence classes replaced by
tolerance classes, introduces in the same manner, a closure operator
$cl_{J}$ lacking the idempotence property
$$
cl_{J}\circ cl_{J} = cl_{J},
$$
nevertheless, the importance of $cl_{J}$ makes it desirable to
study its properties, and it turns out that $cl_{J}$ is a \v{C}ech
topology closure operator.
We give a definition of a \v{C}ech topology. Our presentation
of \v{C}ech topologies is based on the original presentation by
Eduard \v{C}ech in [3].
\begin{deff}
Given a set $X$, we will say that an operator
$$
cl_{\hat{c}}: P(X) \rightarrow P(X),
$$
where $P(X)$ denotes the power set of $X$, is a Cech closure operator,
if it satisfies the following conditions:
\begin{itemize}
\item[$(\hat{c}_{1})$] $cl_{\hat{c}}\phi = \phi$,
\item[$(\hat{c}_{2})$] $(A\subseteq B)\Rightarrow (cl_{\hat{c}}A
\subseteq cl_{\hat{c}} B)$,
\item[$(\hat{c}_{3})$] $A\subseteq cl_{\hat{c}} A$.
\item[] for any $A,B \subseteq X$.
\end{itemize}
It follows from $(\hat{C}_{1}) - (\hat{C}_{3})$ that any \v{C}ech
closure operator is monotone $(\hat{C}_{2})$ and extensive $(\hat{C}_{3})$.
Let us note, that a \v{C}ech closure operator need not satisfy
$$
cl_{\hat{c}}(cl_{\hat{c}}A) = cl_{\hat{c}}A,
$$
in contrast to the classical Kuratowski closure operator.
\end{deff}
\begin{deff}
A pair $(X,cl_{\hat{c}})$, where $X$ is a set and $cl_{\hat{c}}$ is a \v{C}ech
closure operator satisfying $(\hat{C}_{1})-(\hat{C}_{3})$, will be called a
\v{C}ech topological space.
In the sequel, we will use the standard closure symbol $\overline{A}$
to stand for $cl_{\hat{c}}A$; topologies will be denoted, according to
standard notation, with small letters $u,v,\ldots,$ i.e., the
symbol $(X,u)$ will denote a \v{C}ech topological space with a
\v{C}ech topology $u$ - a shortcut for \v{C}ech closure
operator-induced structure.
\end{deff}
\section{Comparison of topologies}
For topologies $u,v$ on a set $X$, we will say that $u$ is
finer than $v$, or $v$ is coarser than~$u$,
$$
\mbox{if:}\;\;\; cl_{u}A\subseteq cl_{v} A \quad\mbox{for any}\quad
A\subseteq X,
$$
where $cl_{u}$ denotes the \v{C}ech
closure operator of the topology $u$, etc.
The finest topology on $X$ is the topology $\overline{u}$ for which
$$
cl_{\overline{u}} A=A\quad \mbox{for}\quad
A\subseteq X,
$$
and the coarsest topology on $X$ is the topology $\underline{u}$ for
which
$$
cl_{\underline{u}} A=X\quad\mbox{for}\quad A\subseteq X, \;\;A\neq \emptyset.
$$
\section{Closed sets}
We will say that a set $A\subseteq X$ is closed if:
$$
\overline{A}=A\quad\mbox{where}\quad
\overline{A}=cl_{u}A
$$
{\it Property 4.3.1.\ $X$ is a closed set.\/}
\vfil\eject
%\medskip
\begin{proof}
Clearly, $X\subseteq \overline{X}$ by $(\hat{c}_{3})$, also $\overline{X}
\subseteq X$, hence
$$
\overline{X}=X.
$$
\end{proof}
{\it Property 4.3.2. \ $\phi$ is a closed set.\/}
\bigskip
\begin{proof}
By $(\hat{c}_{1})$.
{\it Property 4.3.3. \ If every set $A_{i}$ is closed, then $\cap_{i\in
I}A_{i}$ is closed.\/}
\end{proof}
\medskip
\begin{proof}
We have $\cap_{i} A_{i}\subseteq A_{i}$ for each $i\in I$,
hence, by $(\hat{c}_{2}), \overline{\cap A_{i}} \subseteq \overline{A_{i}}$
for $i\in I$ and so
$$
\overline{\cap_{i} A_{i}} \subseteq \cap_{i} \overline{A_{i}}.
$$
As $\overline{A}_{i}=A_{i}$ for $i\in I$, we have
$$
\overline{\cap_{i}A_{i}} \subseteq \cap_{i} A_{i},
$$
hence by $(\hat{c}_{3})$,
$$
\overline{\cap_{i} A_{i}} = \cap_{i}A_{i}.
$$
\end{proof}
Let us remark, that we do not have as a rule, that the
union of two closed sets is a closed set, actually it is not universally
true.
\section{Open sets}
We proceed with a definition of an open set. As usual, it will
be defined as a complement of a closed set.
\begin{deff}
A set $A\subseteq X$ is open iff $X\back A$ is closed.
Thus: ($A$ is open) $\Leftrightarrow \overline{X\back A} =
X\back A \Leftrightarrow$
$$
A=X \back \overline{X\back A}.
$$
Let us restate the duals of 4.3.1 - 4.3.3.
\medskip
{\it Property 1.5.2. \ The set $X$ is open.\/}
\medskip
{\it Property 1.5.3. \ The set $\phi$ is open.\/}
\medskip
{\it Property 1.5.4. \ If $A_{i}$ open for each $i\in I$, then
$\cup_{i} A_{i}$ is open.\/}
\end{deff}
\section{The interior operator}
The interior operator, $Int_{u}$ in a given \v{C}ech
topology $u$ on a set $X$ will be introduced as a dual to
the corresponding closure operator viz.
$$
Int A=X - \overline{X-A}\quad\mbox{for}\quad
A\subseteq X.
$$
Let us put the resulting dual properties of $Int$.
\begin{itemize}
\begin{itemize}
\item[$(Int_{1})$] $Int X=X$,
\item[$(Int_{2})$] $A\subseteq B\Rightarrow Int A\subseteq Int B$,
\item[$(Int_{3})$] $Int A\subseteq A$.
\end{itemize}
\end{itemize}
The $Int$ operator is therefore monotone $(Int_{2})$ and anti-extensive
$(Int_{3})$.
\end{document}