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\title{Math Studies Homework 2}
\author{David Handron}

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    Submit the \LaTeX\ file containing your solutions to the
    email address analysishw@legba.math.cmu.edu by
    class time on Fri 17 September.   

 


\begin{enumerate}

\item Here is an alternative definition of {\em connected}:


\textbf{Definition:} Let $X$ be a metric space.  A {\em separation} of
$X$ is a pair $U,V$ of disjoint nonempty open subsets of $X$ whose
union is $X$.  The space $X$ is {\em connected} if there does not
exist a separation of $X$.


Show that this definition is equivalent to Definition 2.45 in Rudin.

\item  Let $\{A_n\}$ be a sequence of connected subsets of a metric
  space $X$, such that $A_n\cap A_{n+1}\neq\emptyset$ for all $n$.
  Show that $\cup_n A_n$ is connected.

\item Let $A\subset X$.  Show that if $C$ is a connected subspace of
  $X$ that intersects both $A$ and $A^c$ then $C$ contains a point of
  $\overline{A}\cap \overline{A^c}$.

\item If $A$ is a connected subspace of a metric space $X$, does it
  follow that $A^\circ$ is connected?  Does the converse hold?

\item Rudin, p. 78, \#1.

\item Rudin, p. 78, \#2.

\end{enumerate}

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