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\title[Topology Proceedings Example Article]%
{Topology Proceedings \\Example for the Authors}
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\author{Author One}
\address{Department of Mathematics \& Statistics; Auburn University;
Auburn, Alabama 36849}
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%\curraddr{}
\email{topolog@auburn.edu}
%\thanks{The first author was supported in part by NSF Grant \#000000.}
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\keywords{Some objects, some conditions}
\thanks {ALL references are real and correct; ALL citations are imaginary.}
\begin{abstract} This paper
contains a sample article in the Topology Proceedings format.
\end{abstract}
\maketitle
\section{\bf Introduction}
This is a sample article in the TOPOLOGY PROCEEDINGS format.
Prepare your paper in a similar manner before submitting it
to TOPOLOGY PROCEEDINGS.
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parameters like for example\newline
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\section{\bf Main Results}
Let $\mathcal{S}$ denote the set of objects satisfying some condition.
\begin{definition}Let $n$ be a positive integer. An object has
the property $P(n)$ if
some additional condition involving the integer $n$ is satisfied.
We will denote
by $S_n$ the set of all $s$ in $\mathcal{S}$ with
the property $P(n)$.
\end{definition}
The following proposition is a simple consequence of the definition.
\begin{proposition}\label{Prop1}
The sets $S_1,S_2,\dots$ are mutually
exclusive.
\end{proposition}
\begin{lemma}
If $\mathcal{S}$ is infinite then $\mathcal{S}=\bigcup_{n=1}^{\infty}S_n$.
\end{lemma}
\begin{proof}
Since $\mathcal{S}$ is the set of objects satisfying some condition,
it follows from \cite{A}
that
\begin{equation}\label{myeq}
\operatorname{obj}(\mathcal{S})<1.
\end{equation}
By \cite[Theorem 3.17]{E}, we have
\[
\operatorname{obj}(S_n)>2^{-n}
\]
for each positive integer $n$. This result, combined with (\ref{myeq}) and
Proposition \ref{Prop1}, completes the proof of the lemma.
\end{proof}
\begin{theorem}[Main Theorem]
Let $f:\mathcal{S}\to\mathcal{S}$ be a function such that
$f(S_n)\subset S_{n+1}$ for each positive integer $n$. Then the following
conditions are equivalent.
\begin{enumerate}
\item $\mathcal{S}=\emptyset$.
\item $S_n=\emptyset$ for each positive integer $n$.
\item $f(\mathcal{S})=\mathcal{S}$.
\end{enumerate}
\end{theorem}
\begin{remark} Observe that the condition in the definition
of $\mathcal{S}$ may be replaced by some other condition.
\end{remark}
\bibliographystyle{plain}
\begin{thebibliography}{10}
\smallskip
\bibitem{A} A. V. Arhangel'ski\u{i} and Scotty L. Thompson, {\it The cleavability approach to comparing topological spaces}, Questions Answers Gen. Topology {\bf 28} (2010), no. 2, 133--145.
\smallskip
\bibitem{B} Karol Borsuk, {\it On a new shape invariant}, Topology Proc. {\bf 1} (1976), 1--9.
\smallskip
\bibitem{E} Ryszard Engelking, {\it General Topology}. Translated from the Polish by the author. Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. Warsaw: PWN---Polish Scientific Publishers, 1977.
\smallskip
\bibitem{K} Bronis\l av Knaster, {\it On applications of mathematical logic to mathematics} (Czech), \v{C}asopis P\v{e}st. Mat. {\bf 76} (1951), 3--22.
\smallskip
\bibitem{M} Kiiti Morita and Jun-iti Nagata, eds. {\it Topics in General Topology}. North-Holland Mathematical Library, 41. Amsterdam: North-Holland Publishing Co., 1989.
\smallskip
\bibitem{R} Mary Ellen Rudin, {\it A biconnected set in the plane}, Topology Appl. {\bf 66} (1995), no. 1, 41--48.
\smallskip
\bibitem{T} William P. Thurston, {\it On the geometry and dynamics of iterated rational maps}, in Complex Dynamics: Families and Friends. Ed. Dierk Schleicher. Wellesley, MA: A K Peters, 2009. 3--137.
\end{thebibliography}
\end{document}