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\pagestyle{fancy} \lhead{{\sf Homework 7 $|$ Due October 17 (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}

%f03.7
\item A topological space $X$ is \emph{disconnected} if it is the
  union of two disjoint nonempty open sets. Otherwise it is
  \emph{connected}.
  \begin{enumerate}
  \item Prove that $X$ is disconnected if and only if there exists a
    continuous surjective map $f\colon X\to\{0,1\}$, where $\{0,1\}$
    has the discrete topology.
  \item Suppose $X$ has the property that every two point subset lies
    inside a connected subset of $X$. Prove that $X$ is connected. \\
  \end{enumerate}



\item Two subsets $A,B\subseteq X$ are \emph{separated} in $X$ if
  $A\cap\overline{B}=\overline{A}\cap B=\emptyset$. A subspace
  $Y\subseteq X$ is connected if $Y$ is not the union of two separated
  sets. Prove that the following are equivalent:
  \begin{enumerate}[(i)]
  \item $Y$ is separated.
  \item $Y$ is the union of two sets that are open in the subspace
    topology of $Y$.
  \item $Y$ is the union of two sets that are closed in the subspace
    topology of $Y$.  \\
  \end{enumerate}



%f03.7
\item Prove that if $A\subseteq X$ is connected, and $A\subseteq
  B\subseteq\overline{A}$, then $B$ is connected. \\

  

%f97.1
\item A topological space $X$ is \emph{step connected} if given any
  open cover $\mathcal{U}$ of $X$ and any pair of points $x,y\in X$,
  there is a finite sequence $U_1,\dots,U_n$ of sets belonging to
  $\mathcal{U}$ so that $x\in U_1$, $y\in U_n$ and $U_i\cap
  U_{i+1}\neq\emptyset$ of $1\leq j<n$. Prove that $X$ is step
  connected if and only if it is connected. \\


% m24.8
\item A space $X$ is \emph{path connected} if every two points $x,y\in
  X$ can be connected via a \emph{path}. That is, there is a
  continuous function $f\colon[a,b]\to X$ with $f(a)=x$, $f(b)=y$, and
  $f([a,b])\subseteq X$. Prove or disprove the following.
\begin{enumerate}
  \item $X$ and $Y$ are path connected if and only if $X\times Y$ is
    path connected.
  \item If $A\subseteq X$ and $A$ is path connected, then
    $\overline{A}$ is path connected.
  \item If $f\colon X\to Y$ is continuous and $X$ is path connected,
    then $f(X)$ is path connected.
  \item If $\{A_\alpha\}$ is a collection of path connected subspaces
    of $X$ that has a nonempty intersection, then $\cup A_\alpha$ is
    path connected. \\
\end{enumerate}


\end{enumerate}

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