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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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\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.
The article includes two examples of figures with PDF graphics.
\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.
Please do not change the page size and do not redefine the other pagestyle
parameters like for example\newline
\noindent$\backslash$pagenumbering,
$\backslash$pagestyle,
$\backslash$baselineskip,
etc.
\section{\bf Including Figures}
This version of the article includes two examples of figures with PDF graphics. Figure~\ref{tpmap} shows how to include a PDF graphic in the figure environment. Figure~\ref{tpmapwithletters} shows how to add \LaTeX{} lettering and symbols to a graphic. Please see the source file for more information about the figures.
When including graphics please use the following guidelines:
\noindent$\bullet$ We should be able to process your source files either by latex and then by a dvi-to-pdf converter (like dvipdfm), or by pdflatex.
\noindent$\bullet$ Put your graphics in the figure environment and let them float (be positioned automatically within the paper, \LaTeX{} default).
\noindent$\bullet$ Your figures cannot be wider than the standard text width in the paper.
\noindent$\bullet$ Whenever possible, use vector graphics ("drawings" with objects geometrically defined). If bitmap graphics must be used (for example, if a photograph is included), please make sure that the resolution is high enough to look good when printed on a 600 dpi black and white printer.
\noindent$\bullet$ If your figure contains lettering, its fonts, sizes and styles must match those of the body of the articles. This is especially important for all math lettering and symbols. It is usually easier to achieve the proper \LaTeX{} standard if the lettering is added in \LaTeX{}, see Figure~\ref{tpmapwithletters} for an example.
\noindent$\bullet$ Color may be used in graphics, but the figures should also be readable when printed on a 600 dpi black and white printer. (The print version of TOPOLOGY PROCEEDINGS is in black and white only, while the electronic version will show color if used.)
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\begin{figure}
\begin{picture}(260,71)
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\put(125,40){$f_{\epsilon}$}
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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}