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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due November 7 (Thursday)
}} \rhead{\thepage} \cfoot{{\sf MthSc 4560/6560 $|$ Topology $|$ Fall
    2013 $|$ M.~Macauley}}



\begin{document}

$\;$

\noindent Throughout, a topological space $X$ is endowed with a
topology $\tau$, even if not explicitly mentioned.

\begin{enumerate}

%m51.1
\item Show that if $f,f'\colon X\to Y$ are homotopic maps (not
  necessarily paths) and $g,g'\colon Y\to Z$ are homotopic, then
  $g\circ f$ and $g'\circ f'$ are homotopic. \\



%m51.3
\item Given spaces $X$ and $Y$, let $[X,Y]$ denote the set of homotopy
  classes of maps of $X$ into $Y$. A space $X$ is \emph{contractible}
  if the identity map $i_X\colon X\to X$ is nullhomotopic.
  \begin{enumerate}
    \item Show that $I$ and $\R$ are contractible. 
    \item Show that a contractiable space is path connected.
    \item Show that if $Y$ is contractible, then for any $X$, the set
      $[X,Y]$ has a single element.
    \item Show that if $X$ is contractible and $Y$ is path connected,
      then $[X,Y]$ has a single element. \\
  \end{enumerate}



\item Let $f\colon I\to X$ be a path with $f(0)=x_0$ and $f(1)=x_1$.
  Denote its path homotopy class by $[f\,]$, and let $\bar{f}\colon I\to
  X$ be the reverse path: $\bar{f}(s)=f(1-s)$. Let $e_{x_1}\colon I\to
  X$ denote the constant path at $x_1$.
  \begin{enumerate}
  \item Show that $[f\,]*[e_{x_1}]=[f\,]$.
  \item Show that $[\bar{f}\,]*[f\,]=[e_{x_1}]$.
  \end{enumerate}
  


\end{enumerate}

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