Instead of providing headings manually (which can be prone-error and implies a lot of manual intervention), you can redefine the \part command to give the desired layout; then you can make the section counter reset every time part is incremented:
\documentclass[12pt]{article}
\makeatletter
\@addtoreset{section}{part}
\def\@part[#1]#2{%
\ifnum \c@secnumdepth >\m@ne
\refstepcounter{part}%
\addcontentsline{toc}{part}{\thepart\hspace{1em}#1}%
\else
\addcontentsline{toc}{part}{#1}%
\fi
{\parindent \z@ \raggedright
\interlinepenalty \@M
\normalfont\centering
\ifnum \c@secnumdepth >\m@ne
\LARGE\bfseries \partname\nobreakspace\thepart
\par\nobreak
\fi
\huge \bfseries #2%
\markboth{}{}\par}%
\nobreak
\vskip 3ex
\@afterheading}
\renewcommand\partname{Topic}
\makeatother
\begin{document}
\part{}
\section{Section I}
content
\section{Section II}
content
\subsection{Subsection I}
content
\subsection{Subsection II}
content
\section{Section III}
content
\part{}
\section{Section I}
content
\section{Section II}
content
\subsection{Subsection I}
content
\subsection{Subsection II}
content
\section{Section III}
content
\end{document}

Adendum:
Using the xpatch package you can simplify the code:
\documentclass[12pt]{article}
\usepackage{xpatch}
\makeatletter
\@addtoreset{section}{part}
\xpatchcmd{\@part}{\normalfont}{\normalfont\centering}{}{}
\xpatchcmd{\@part}{\Large}{\LARGE}{}{}
\renewcommand\partname{Topic}
\makeatother
\begin{document}
\part{}
\section{Section I}
content
\section{Section II}
content
\subsection{Subsection I}
content
\subsection{Subsection II}
content
\section{Section III}
content
\part{}
\section{Section I}
content
\section{Section II}
content
\subsection{Subsection I}
content
\subsection{Subsection II}
content
\section{Section III}
content
\end{document}
Adendum 2:
Another variant, using this time the titlesec package to easily customize the \part command, suppressing the "Part #" label and centering the title:
\documentclass[12pt]{article}
\usepackage{titlesec}
\makeatletter
\@addtoreset{section}{part}
\makeatother
\titleformat{\part}[display]
{\normalfont\LARGE\bfseries\centering}{}{0pt}{}
\begin{document}
\part{Polynomials}
\section{Section I}
content
\section{Section II}
content
\subsection{Subsection I}
content
\subsection{Subsection II}
content
\section{Section III}
content
\part{Functions}
\section{Section I}
content
\section{Section II}
content
\subsection{Subsection I}
content
\subsection{Subsection II}
content
\section{Section III}
content
\end{document}
