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\title
{Topology Notes \\ Summer, 2006}

\author[J.~C.~Mayer]{John C.~Mayer} \email[John C.~Mayer]{mayer@math.uab.edu}

\address[John C.~Mayer]
{Department of Mathematics\\ University of Alabama at Birmingham\\
Birmingham, AL 35294-1170}


\keywords{topology} \subjclass{Primary: 54F20}
\date{\today}

\maketitle

\section{Connected Spaces}

One of the big ideas in topology, motivated by its usefulness in
analysis (think ``intermediate value theorem"), is the idea of {\em
connectedness}. We are trying to capture the idea that $X$ is/is not
broken into ``pieces."  See Munkries, Section 23.

\begin{defn} \label{connected} Let $X$ be a space.  We say that $X$ is {\em
separated} iff there exist nonempty open sets $U$ and $V$ in $X$
such that $X=U\cup V$ and $U\cap V=\0$.  If there are such open
sets, then we call $X=U\cup V$ a {\em separation} of $X$.  We say
$X$ is {\em connected} if there does not exist a separation of
$X$.
\end{defn}

\begin{exr} On $X=\{0,1,2\}$, (1) give an example of a nontrivial topology such
that $X$ is connected, and (2) give an example such that $X$ is
not connected.
\end{exr}

\begin{prop} Let $X$ be a space.  Then $X$ is connected iff the
only subsets of $X$ that are both open and closed are $\0$ and $X$.
\end{prop}

\begin{prob} Let $\mc T_1$ and $\mc T_2$ be topologies on the set
$X$.  Suppose $\mc T_1\subset\mc T_2$.  What does connectedness of
$X$ in one topology imply about connectedness of $X$ in the other
topology? \end{prob}

We also want to be able to talk about {\em connected subsets} of a
space.

\begin{defn}  Let $X$ be a space and $Y\subset X$.  We say that
$Y$ is a {\em connected subset} of $X$ iff $Y$ is connected in the
subspace topology. \end{defn}

\begin{prop}  Let $X$ be a space and $Y\subset X$. Then $Y$ is
separated iff there exist nonempty sets $A$ and $B$ in $Y$ such that
$Y=A\cup B$, $A\cap B=\0$, and neither $A$ nor $B$ contains a limit
point of the other. $\Box$ \end{prop}

\begin{exm} Give $\nat$ the natural order and the order topology.  Show that
$\nat$ is not connected.  What are the connected subsets of $\nat$?
\end{exm}

\begin{prob} Give $\nat$ the cofinite topology.  Is $\nat$ connected in the cofinite
topology? \end{prob}

\begin{prop} Let $X=U\cup V$ be a separation of $X$.  Suppose $Y$
is a connected subset of $X$.  Then either $Y\subset U$ or
$Y\subset V$. $\Box$ \end{prop}

\begin{prop} Let $X$ be a space and $\mc C$ a collection of
connected subsets of $X$ with the property that $\exists x\in X$
such that $\forall C\in \mc C, x\in C$. Then $\cup \mc C$ is
connected. $\Box$
\end{prop}

\begin{prop} Let $X$ be a space and $C$ a connected subset of $X$.
 Suppose $C\subset D\subset \cl C$.  Then $D$ is connected.  $\Box$
 \end{prop}

\begin{thm} The continuous image of a connected set is connected. $\Box$
\end{thm}

\begin{thm} The finite Cartesian product of connected spaces is
connected. $\Box$ \end{thm}

\begin{exm} $\real^\omega$ is connected in the product toplogy, but
not connected in the box topology.  $\Box$ \end{exm}

\subsection*{Assignment 1}
This assignment is due Friday, June 9.

  Easy - 1--5: The ``unboxed" items
above.

Harder - 6--9: Page 152: 2, 6, 7, 11

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