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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due September 26 (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}


\item Suppose $(X,\tau_X)$ and $(Y,\tau_Y)$ are topological spaces and
  $A\subseteq X$ and $B\subseteq Y$. Prove that:
  \begin{enumerate}
  \item $\Int(A\times B)=\Int(A)\times\Int(B)$;
  \item $\overline{A\times B}=\overline{A}\times\overline{B}$;
  \item $\partial(A\times B)=[\partial A\times\overline{B}]\cup
    [\overline{A}\times\partial B]$. \\
  \end{enumerate}


\item Let $X_1,X_2$ be sets and let $\pi_i\colon X_1\times X_2\to X_i$
  denote the standard projection maps for $i=1,2$. The Cartesian product
  $X_1\times X_2$ can be thought of as the ``smallest'' set that
  projects onto both $X_1$ and $X_2$. This can be formalized by the
  following ``universal property'': \\

  For any set $Z$ along with maps $f_i\colon Z\to X_i$, there is a
  unique map $f\colon Z\to X_1\times X_2$ satisfying $\pi_i\circ
  f=f_i$ for $i=1,2$. That is, the following diagram commutes:
  \[
  \xymatrix{ & Z \ar@{-->}[d]^{\exists ! f}\ar[dl]_{f_1}\ar[dr]^{f_2} & \\
    X_1 & X_1\times X_2\ar[l]^{\pi_1}\ar[r]_{\pi_2} & X_2}
  \]
  Prove this. \\


  
\item Let $(X_1,\tau_1),\dots,(X_n,\tau_n)$ be topological spaces and
  endow $X:=\prod_{i=1}^n X_i$ with the product topology $\tau_X$. Let
  $\pi_i\colon X\to X_i$ be the standard projection maps; recall these
  are continuous.
  \begin{enumerate}
  \item Suppose $f\colon Z\to X$ is a function from a topological
    space $(Z,\tau_Z)$. Show that $f$ is continuous if and only if
    each $f_i:=\pi_i\circ f$ is continuous:
    \[
    \xymatrix{X\ar[r]^{\pi_i} & X_i \\ Z\ar[u]^f\ar[ur]_{\pi_i\circ f} }
    \]
  \item Prove the following ``universal property'' of the product
    topology: For any collection of continuous maps $f_i\colon Z\to
    X_i$, there exists a unique continuous map $f\colon Z\to X$ such
    that $f_i=\pi_i\circ f$ for all $i$. That is, the following
    diagram commutes:
    \[
    \xymatrix{X\ar[r]^{\pi_i} & X_i \\ Z\ar@{-->}[u]^f\ar[ur]_{f_i} }
    \]
  \end{enumerate}


%\item Let $S$ be a nonempty set and $X=2^S$, the power set of $S$. Define 
%  \[
%  \tau=\{U\subseteq S:|S\setminus U|<\infty\}\cup\{\emptyset\}.
%  \]
%  \begin{enumerate}
%    \item Show that $\tau$ is a topology on $X$. 
%    \item Define $f\colon X\times X\to X$ by $f(A,B)=A\cap B$. Prove
%      that $f$ is continuous.
%    \item Define $g\colon X\times X\to X$ by $g(A,B)=A\cup
%      B$. Formulate (with proof) a necessary and sufficient condition
%      on $S$ that ensures that $g$ is continuous.
%  \end{enumerate}


\end{enumerate}

\end{document}
