Topology Proceedings Template
作者:
Steven Clontz
最近上传:
18 小时前
许可:
Creative Commons CC BY 4.0
摘要:
Official template required for submission to Topology Proceedings.
\begin
Discover why over 20 million people worldwide trust Overleaf with their work.
Official template required for submission to Topology Proceedings.
\begin
Discover why over 20 million people worldwide trust Overleaf with their work.
% Official template for
% https://topology.journals.yorku.ca/index.php/tp/about/submissions
% Version 2025-04-25
\documentclass{amsart}
\usepackage{topproc}
% Author info
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% First author:
\author{Author One}
\address{Department of Mathematics \& Statistics; Auburn University;
Auburn, Alabama 36849}
% Current address (if needed):
%\curraddr{}
\email{topolog@auburn.edu}
% \thanks{The first author was supported in part by NSF Grant \#000000.}
% Second author (if needed):
%\author{Author Two}
%\address{}
%\email{}
%\thanks{Support information for the second author.}
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% Title
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\title[Topology Proceedings Example Article]%
{Topology Proceedings \\Example for the Authors}
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% General info
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\subjclass[2020]{Primary 54X10, 58Y30, 18D35; Secondary 55Z10}
%
% Please use the current 2020 Mathematics Subject Classification:
% https://mathscinet.ams.org/mathscinet/msc/msc2020.html
% https://zbmath.org/classification/
\keywords{Some objects, some conditions}
\thanks {ALL references are real and correct; ALL citations are imaginary.}
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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.
We require that your paper be typeset in \LaTeX{} using this
approved format in order to be considered for publication in
Topology Proceedings.
Please do not change the page size and do not redefine any
parameters such as \verb|\pagenumbering|,
\verb|\pagestyle|, \verb|baselineskip|, etc. Please do not
disable the automatic line numbering as these numbers will be
used by referees in their report.
\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}