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\title{Homework 6 for 18.100B/C, Section 1, Fall 2010\\
Due Thursday 28 October by 11AM in class or 2-108\\
or by email before 5PM.}

\maketitle

\begin{enumerate}
\item Rudin Chap 4, No 2. If $f:X\longrightarrow Y$ is a continuous map
between metric spaces, $E\subset X$ and $f(E)\subset
  Y$ is its image under $f,$ show that the closures satisfy 
%
\begin{equation*}
f(\overline{E})\subset\overline{f(E)}.
\label{BASE.1}\end{equation*}
Give an example where the right side is strictly larger than the left.
%
\item Consider the cartesian product $X\times Y=\{(x,y);x\in X,\ y\in Y\}$
  of two metric spaces with the distance
%
\begin{equation*}
D((x,y),(x',y'))=d_X(x,x')+d_Y(y,y').
\label{BASE.4}\end{equation*}
If $f:X\longrightarrow Y$ is any map, define its graph by
\begin{equation*}
G(f)=\{(x,y)\in X\times Y;x\in X,\ y=f(x)\}.
\label{BASE.2}\end{equation*}
%
\subitem Show that if $f$ is continuous then $G(f)$ is closed.
\subitem Show that if $X$ is compact then a map $f:X\longrightarrow Y$ is
continuous if and only if its graph is compact.
%
\item Let $X_1$ and $X_2$ be closed subsets of a metric space $X$ such that
  $X=X_1\cup X_2$ and suppose $g_i:X_i\longrightarrow Y,$ $i=1,2,$ are two
  continuous maps defined on them. Show that if $g_1(x)=g_2(x)$ for all
  $x\in X_1\cap X_2$ then $g:X\longrightarrow Y$ where $g(x)=g_i(x)$ for
  $x\in X_i$ is continuous. 
\item Let $\{y_n\}$ be a sequence in a metric space $Y.$ Define a function
  on the set 
%
\begin{equation*}
D=\{1/n\in[0,1];n\in\mathbb{N}\}
\label{BASE.3}\end{equation*}
%
by $f(\frac1n)=y_n.$ Show that $f$ has a limit at $0$ if and only if
  $\{y_n\}$ is convergent.
\item Rudin Chap 4, No. 14. Show that any continuous map
  $f:[0,1]\longrightarrow [0,1]$ must have a fixed point, that is there
  exists at least one point $x\in[0,1]$ such that $f(x)=x.$
\end{enumerate}

Here are some questions on connected sets, designed to clarify things a
little. They are for your amusement only.


\begin{enumerate}
\item Recall that given a subset $E\subset X$ of a metric space we have
  defined the condition on a subset $F\subset E$ that it be relatively open
  (or relatively closed) and the characterization of this. Check that
  $F\subset E$ is relatively closed in $E$ if and only if
  $F=\overline{F}\cap E$ where $\overline{F}$ is the closure in $X.$ Show
  that a subset $E\subset X$ is connected if and only if the only decompositions
  of it into two disjoint relatively closed subsets $E=A\cup B$ has one of
  the sets empty.
\item Suppose $f:X\longrightarrow Y$ is continuous and $E\subset X,$ show
  that $f\big|_E:E\longrightarrow Y$ is continuous with the metric on $E$
  induced from $X.$
\item Show that if and $f:X\longrightarrow Y$ is continuous and $U\subset
  f(X)$ is relatively open (resp. relatively closed) set then $f^{-1}(U)$
  is open (resp. closed).
\item Suppose $E\subset X$ is connected and $f:X\longrightarrow Y$ is
  continuous, show that if $f(E)=A\cup B$ is a decomposition into
  relatively closed subsets then $E=(E\cap f^{-1}(A))\cup (E\cap
  f^{-1}(B))=f\big|_E^{-1}(A)\cup f\big|_E^{-1}(B)$
  is a decomposition into relatively closed subsets.
\item Deduce from this that the continuous image of a connected set is connected.
\end{enumerate}


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