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\title{Topology}
\author{Scot Free Kennedy}
\maketitle

\section{Main Idea}

A topology on a set ${\mathcal S}$ is, in essence, a ``way to talk about
subsets
of ${\mathcal S}$''. That is, it is a system of subsets which is, in some
sense,
well-behaved: we will be able to do what we wish with the sets. Note that
there are many types of things we would like to do with subsets of any
particular set of interest; thus we will want to be able to talk about
subsets in many different ways; thus there are many possible topologies on
any one set.

Before getting in to details, let's look at a few examples:
\begin{description}
\item{Discrete} This is simply the set of all possible subsets of
${\mathcal S}$.
\item{Indiscrete} This is the most ``coarse'' topology, consisting of
$\{ {\mathcal S}, \emptyset \}$, that is, the entire set and the empty
set.
\item{Standard Toplogy on the Reals} This is the familiar topology on the
Real line, generated by all possible open intervals.
\item{Lower Limit Topology on the Reals} This is generated by
``half-open'' intervals, ie, sets of the form $[a,b)$.

\end{description}


\section{Basic Theory and Definitions}


\section{Connections}


\section{Literature}




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