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\includepdf{18-327-FinalFASCover.pdf}

\noindent{\red Red - post factum comments.

\vfill\noindent General. Point reductions, though mild, for irrelevant nonsense.}

\vfill\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'' {\red(relative to the standard topology)}. \qquad{\red\footnotesize (4 points)}
\item Prove that CC$\Rightarrow$MC. \qquad{\red\footnotesize (8 points)}
\item Prove that MC$\Rightarrow$CC. \qquad{\red\footnotesize (8 points)}
\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.}

\noindent{\red\footnotesize
{\bf Q.} Do I need to prove that $d$ itself is a continuous function? {\bf A.} No.

\noindent Marking: (4) Being oriented \qquad(4) Continuous \qquad(6) 1-1 \qquad(6) Continuous inverse.

\noindent (-1) In proving that $\phi$ is continuous, used the irrelevant fact that $\pi_y$ is continuous.

\noindent (-4) To show that $\phi$ is open, showed that for al $y$, $\phi_y$ is open.
}

\vfill\noindent{\bf Problem 3.} Let $U$ be a subset of $\bbR^2$.
\begin{enumerate}[itemsep=0mm]
\item Define ``$U$ is connected''. \qquad{\red\footnotesize (4 points)}
\item Define ``$U$ is path-connected''. \qquad{\red\footnotesize (4 points)}
\item Show that if $U$ is open and connected then it is path-connected. \qquad{\red\footnotesize (12 points)}
\end{enumerate}

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

\noindent{\red\footnotesize
$(-2)$ in 1, omitted $A\cup B=U$ and $A\neq\emptyset$ and $B\neq\emptyset$.
\par\noindent $(-1)$ in 3, omitted $A\neq\emptyset$.
\par\noindent $(-2)$ in 3, correct argument but no ``$B_\epsilon(x)\subset U$'' references.
\par\noindent $(-4)$ Proved that $A$ is closed but not open.
}

\newpage\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).

\noindent{\red\footnotesize
(2/20) Did only the case of $F=A\times B$.
\par\noindent (2/20) ``$\pi_1$ is open therefore it is closed''.
}

\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 $\alpha_2$. {\footnotesize\red[``or'' is slightly ambiguous] \qquad (8 points)}
\item Prove that if $X$ is $T_3$ and $\alpha_2$, then it is also $T_4$. \qquad{\red\footnotesize (12 points)}
\end{enumerate}

\noindent{\red\footnotesize
(-2) Correctly defined $T_{3.5}$, instead of $T_3$.
\par\noindent (-4) $U'_n\coloneqq U_n\setminus\bigcup_{k\leq n}V_k$, instead of $U'_n\coloneqq U_n\setminus\bigcup_{k\leq n}\overline{V_k}$.
\par\noindent (-11) Used Urysohn metrization.
\par\noindent (-11) A countable intersection is open.
\par\noindent (-8) ``Half'' of the unfriending construction.
}

\vfill\noindent{\bf Problem 6.} Let $(X,d)$ be a metric space.
\begin{enumerate}[itemsep=0mm]
\item Define ``$X$ is complete''. \qquad{\red\footnotesize (4 points)}
\item Suppose that for some $\epsilon>0$, every $\epsilon$-ball in $X$ has a compact closure. Show that $X$ is complete. \qquad{\red\footnotesize (8 points)}
\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. \qquad{\red\footnotesize (8 points)}
\end{enumerate}

\noindent{\red\footnotesize
{\bf Q.} Can I use that compact plus metric implies complete? {\bf A.} Yes.
}

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

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