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\address{Department of Mathematics \& Statistics; Auburn University;
Auburn, Alabama 36849}
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\email{topolog@auburn.edu}
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\title[Topology Proceedings Example Article]%
{Topology Proceedings \\Example for the Authors}
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\keywords{Some objects, some conditions}
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\begin{document}
\topProcLogo
\begin{abstract}
This paper contains a sample article in the
Topology Proceedings format.
\end{abstract}
\maketitle
\section{Introduction}
This is a sample article in the Topology Proceedings format.
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\section{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{zbMATH05917775}
that
\begin{equation}\label{myeq}
\operatorname{obj}(\mathcal{S})<1.
\end{equation}
By \cite[Theorem 3.17]{zbMATH00042114}, 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
$f(S_n)\subseteq S_{n+1}$ 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}
\printbibliography
\end{document}