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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}

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\section{Real Line}

We know a point is connected, and we know an indiscreet space is
connected.  But that about exhausts our current state of examples.
In this section we prove a general theorem about a kind of ordered
space from which is follows that the real line and any of its
intervals or rays is connected.  (See Munkries, Section 24.)  The
key properties that make this possible are in the following
definition.

\begin{defn}  A nondegenerate simply ordered space $L$ is called a {\em
linear continuum} iff it satisfies the following:
\begin{enumerate}
\item $L$ has the least upper bound property.
\item If $x,y$ in $L$, then there exists $z\in L$ such that $x<z<y$.
\end{enumerate}
A subset $C$ of an ordered set is {\em convex} iff whenever $a,b\in
C$, $a\le b$, then the interval $[a,b]\subset C$.
\end{defn}

\begin{thm} If $L$ is a linear continuum, then $L$ is connected, and
so is any convex set in $L$. $\Box$ \end{thm}

\begin{cor} The real line $\real$ is connected, and so are rays and
intervals in $\real$. $\Box$ \end{cor}

\begin{cor} [Intermediate Value Theorem] Let $f:X\to Y$ be a
continuous function of a connected space into an ordered space.  If
for $a,b\in X$, $r\in Y$ lies between $f(a)$ and $f(b)$ in the order
on $Y$, then there is a point $c\in X$ such that $f(c)=r$. $\Box$
\end{cor}

\begin{exm} Let $X$ be a well-ordered set.  Then $X\times[0,1)$ in
the dictionary order topology is a linear continuum. $\Box$
\end{exm}

\begin{exm} In particular, $\nat\times[0,1)$ is homeomorphic to the real
ray $[0,\infty)$.
\end{exm}

\begin{prop} Let $f:[0,1] \to [0,1]$ be continuous.  Then there is a
{\em fixed point} of $f$; that is, a point $x\in[0,1]$ such that
$f(x)=x$. $\Box$ \end{prop}

Intuitively, two things are connected to each other if you can get
from one to the other by a path with no breaks.  We make this idea
formal.

\begin{defn} Let $X$ be a space and $x,y\in X$.  We say a
function $f:[a,b]\to X$ is a {\em path} from $x$ to $y$ iff $f$ is
continuous, $f(a)=x$, and $f(b)=y$.  If moreover, $f$ is one-to-one,
we say $f$ is an {\em arc}.  A space is said to be {\em path
connected} (respectively, {\em arcwise connected}) iff every two
(different) points of $X$ can be connected by a path (arc).
\end{defn}

\begin{prop} A path connected space is connected. \end{prop}

\begin{exm}  Connected does not imply path connected, as the following
examples illustrate. \begin{enumerate} \item The square $I\times I$
(where $I=[0,1]$) with the dictionary order topology is connected
but not path connected.
\item The {\em sin 1/x continuum} $X=\cl{\{(x,y)\in\real^2\mid
0<x\le 1 \text{ and } y=\sin(\frac1x)\}}$ is connected but not path
connected.
\end{enumerate} $\Box$
\end{exm}

\section*{Assignment 2, part 1}

Easy:  1--3:  unboxed items above and page 158: 4

Harder:  4--7  : page 152: 8; page 158: 8a, 8c, 10.

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