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UF MTG 4302/5316\\
Introduction to Topology 1\\
Fall 2023
\begin{center}
{\large \bf Homework 5}\\
Due Wednesday, November 1, anytime, on Canvas
\end{center}

{\bf Reading.}
Munkres \S26 -- \S32


{\bf Problems.}
\begin{itemize}

\item \S23 \#9.
Let $A$ be a proper subset of $X$, and let $B$ be a proper subset of $Y$.
If $X$ and $Y$ are connected, show that $(X\times Y) - (A\times B)$ is connected.

\emph{Remark: This problem has appeared on the first year exam before; see for example (\footnote{\url{https://gma.math.ufl.edu/gma/wp-content/uploads/sites/130/FY_Topology_2013_01_part1.pdf}}).}

\item \S24 \#1(a,c).

(a) Show that no two of the spaces $(0,1)$, $(0,1]$, and $[0,1]$ are homeomorphic.\\
\emph{Hint:} What happens if you remove a point from each of these spaces?

(c) Show that $\R^n$ and $\R$ are not homeomorphic if $n>1$.

\item \S26 \#3.
Show that a finite union of compact subspaces of $X$ is compact.

\item \S28 \#7(a,b).
Let $(X,d)$ be a metric space.
If $f$ satisfies the condition $d(f(x),f(y))<d(x,y)$ for all $x,y\in X$ with $x\neq y$, then $f$ is called a \emph{shrinking map}.
If there is a number $\alpha<1$ such that $d(f(x),f(y))\le\alpha d(x,y)$ for all $x,y\in X$, then $f$ is called a \emph{contraction}.
A \emph{fixed point} of $f$ is a point $x$ such that $f(x)=x$.

(a) If $f$ is a contraction and $X$ is compact, show that $f$ has a unique fixed point.

(b) Show more generally that if $f$ is a shrinking map and $X$ is compact, then $f$ has a unique fixed point.

\emph{Remark: This ``Contraction mapping theorem'' problem is on the first year exam syllabus: (\footnote{\url{https://math.ufl.edu/first-year-exam-syllabi/mtg-5316-introduction-to-topology-1/}}).
See our textbook for some hints, though those hints don't need to be followed.}


\end{itemize}

\vfill

{\bf Recommend Problems (not to turn in).}
\begin{itemize}

\item \S25 \#3.

\item \S27 \#4.

\item \S28 \#6.

\end{itemize}

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