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%\noindent{\bf Solve 5 of the following 6 problems. } Each problem is worth 20 points. You have three hours.

\noindent{\bf Problem 1.} For the purpose of this problem, we say that a function $f\colon\bbR^n\to\bbR^m$ is ``Classically Continuous'' (CC) if $\forall x\in\bbR^n\,\forall\epsilon>0\,\exists \delta>0\, \forall y\in\bbR^n\, |x-y|<\delta \Rightarrow |f(x)-f(y)|<\epsilon$, and we say that it is ``Modern-Continuous'' if whenever $U\subset\bbR^m$ is open, $f^{-1}(U)$ is an open subset of $\bbR^n$.
\begin{enumerate}[itemsep=0mm]
\item Define ``$U\subset\bbR^m$ is open''.
\item Prove that CC$\Rightarrow$MC.
\item Prove that MC$\Rightarrow$CC.
\end{enumerate}

\vskip 1mm\par\noindent{\small {\em Tip. } Don't start working! Read the whole exam first. You may wish to start with the questions that are easiest for you.}

\vskip 1mm\par\noindent{\small {\em Tip. } Neatness, cleanliness, and organization count, here and everywhere else!}

\vfill\noindent{\bf Problem 2.} Let $(X,d)$ be a metric space. Define $\phi\colon X\to\bbR^X$ by $\phi(x)_y\coloneqq d(x,y)$, whenever $x,y\in X$. Prove that the map $\phi$ is an embedding (namely, that it is a homeomorphism of $X$ and $\phi(X)$).

\vskip 1mm\par\noindent{\small {\em Tip. } 1-1? Continuous? Continuous inverse?}

\vskip 1mm\par\noindent{\small {\em Tip. } In a ``fresh'' exercise you are welcome to use anything proven in class or in any homework assignment or term test.}

\vfill\noindent{\bf Problem 3.} Let $U$ be a subset of $\bbR^2$.
\begin{enumerate}[itemsep=0mm]
\item Define ``$U$ is connected''.
\item Define ``$U$ is path-connected''.
\item Show that if $U$ is open and connected then it is path-connected.
\end{enumerate}

\par\noindent{\small {\em Hint. } Fix $x_0\in U$ and show that the set of
points in $U$ that can be reached from $x_0$ by a path within $U$ is
clopen.}

\vfill\noindent{\bf Problem 4.} Prove that if $Y$ is a compact topological space and $X$ is an arbitrary topological space then the projection $\pi_1\colon X\times Y\to X$, defined by $(x,y)\mapsto x$, is a closed map (namely, it sends closed sets to closed sets).

\vfill\noindent{\bf Problem 5.} Let $X$ be a $T_1$ topological space.
\begin{enumerate}[itemsep=0mm]
\item Define what it means for $X$ to be $T_3$, $T_4$, or $\alpha_2$.
\item Prove that if $X$ is $T_3$ and $\alpha_2$, then it is also $T_4$.
\end{enumerate}

\vfill\noindent{\bf Problem 6.} Let $(X,d)$ be a metric space.
\begin{enumerate}[itemsep=0mm]
\item Define ``$X$ is complete''.
\item Suppose that for some $\epsilon>0$, every $\epsilon$-ball in $X$ has
a compact closure. Show that $X$ is complete.
\item Suppose that for each $x\in X$ there is an $\epsilon>0$ such that the
ball $B_\epsilon(x)$ has compact closure. Show by means of an example that
$X$ need not be complete.
\end{enumerate}

\vfill \label{goodluck}\centerline{\bf \large Good Luck!}\vfill

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