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\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}

\begin{document}

\title{Analysis II}
\author{Homework 3}
\date{Due in class Monday, February 25, 2019}
\maketitle

\begin{enumerate}

\item Let $X$ be a metric space and $E \subset X$.  We say that $x \in
X$ is a \emph{boundary point} of $E$ if every neighborhood of $x$
contains at least one point in $E$ and at least one point in $E^c$.
The set of all boundary points of $E$ is denoted $\partial E$.
Prove the following.

\begin{enumerate}
\item $E \setminus \partial E$ is open.
\item $E \cup \partial E$ is closed.
\item $\partial E$ is closed.
\end{enumerate}

\item Let $X$ be a set.  Define, for all $x,y \in X$, the ``trivial
metric'' 
\[
  d(x,y) =
  \begin{cases}
    0 & \text{if } x=y \,, \\
    1 & \text{if } x \neq y \,.
  \end{cases}
\]
\begin{enumerate}
\item Show that $d$ is indeed a metric.
\item Show that in this metric space, every subset $E \subset X$ is
both open and closed.
\end{enumerate}

\item Show that the union of a finite number of compact subsets of a
metric space $X$ is compact. 

\item Let $X$ be a metric space and $K \subset X$ compact.  Let
$\{G_\alpha\}$ be an open cover of $K$.  Show that there exists
$\lambda > 0$ such that for every $E \subset K$ with
$\operatorname{diam} E \leq \lambda$ there exists an index $\alpha$
such that $E \subset G_\alpha$.

\item Let $E \subset \R$ be bounded and $f \colon E \to \R$ be
uniformly continuous.  Show that $f$ is bounded on $E$.

\item Let $X,Y$ be metric spaces and $f\colon X \to Y$ continuous.
Show that $f(K)$ is compact if $K$ is compact.


\end{enumerate}





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