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\begin{document}
\noindent{\small
\href{\myurl}{Dror Bar-Natan}:
\href{\myurl/classes/}{Classes}:
\href{\myurl/classes/\#1819}{2018-19}:
\hfill\url{http://drorbn.net/18-327}
}
\begin{center}
\vfill
\noindent{\bf Do not turn this page until instructed.}
\vfill
\href{http://drorbn.net/18-327}{\large MAT327 Introduction to Topology}
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{\LARGE \href{http://drorbn.net/18-327/TT.html}{Term Test}}
\vfill
University of Toronto, October 16, 2018
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\noindent{\bf Solve 4 of the 5 problems on the other side of this page. }\\
Each problem is worth 25 points even though they are not equally difficult.\\
You have an hour and fifty minutes to write this test.
\vfill
\noindent{\bf Notes}
\begin{itemize}
\item No outside material other than stationary is allowed.
\item {\bf Neatness counts! Language counts!} The {\em ideal} written
solution to a problem looks like a page from a textbook; neat and clean
and consisting of complete and grammatical sentences. Definitely phrases like
``there exists'' or ``for every'' cannot be skipped. Lectures are mostly
made of spoken words, and so the blackboard part of proofs given
during lectures often omits or shortens key phrases. The ideal written
solution to a problem does not do that.
\item Do not write on this examination form! Only what you write in the examination booklets counts towards your grade.
\item Indicate clearly which problems you wish to have marked; otherwise an arbitrary subset of the problems you solved will be used.
%{\red \item In red: post-exam additions/notes.}
\end{itemize}
\end{center}
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{\bf Good Luck!}
\end{center}
\newpage
\noindent{\bf Solve 4 of the following 5 problems. } Each problem is worth
25 points. You have an hour and fifty minutes. {\bf Neatness counts! Language counts!}
\vfill\noindent{\bf Problem 1. }
\begin{enumerate}
\item Define the ``finite-complement topology'' on a given set $X$.
\item Let $X$ and $Y$ be sets taken with their finite-complement topologies. Prove that a function $f\colon X\to Y$ is continuous if and only if it is either constant or ``finite to one'' (meaning that $\forall y\in Y,\, |f^{-1}(y)|<\infty$).
\end{enumerate}
\par\noindent{\small {\bf Tip. } Don't start working! Read the whole exam first. You may wish to start with the questions that are easiest for you.}
\vfill\noindent{\bf Problem 2. }
\begin{enumerate}
\item Define ``a topological space $X$ is Hausdorff ($T_2$)''.
\item Prove that a topological space $X$ is Hausdorff if and only if the diagonal $\Delta=\{(x,x)\colon x\in X\}$ is a closed subset of $X\times X$.
\end{enumerate}
\par\noindent{\small {\bf Tip. } ``If and only if'' always means that there are two things to prove.}
\vfill\noindent{\bf Problem 3. } Let $B$ be the set of bounded sequences of real numbers. It is a subset of the set $X=\bbR^\bbN$ of all sequences of real numbers.
\begin{enumerate}
\item Prove that if $X$ is taken with the box topology, then $B$ is both an open and a closed subset.
\item Prove that if $X$ is taken with the product (cylinders) topology, then $B$ is neither open nor closed.
\end{enumerate}
\vfill\noindent{\bf Problem 4. } Given a set $X$ equipped with a metric $d$, prove that there exists a unique topology on $X$ for which the following two properties hold:
\begin{enumerate}
\item For every $x\in X$, the function $f_x\colon X\to\bbR$ defined by $f_x(y)=d(x,y)$ is continuous.
\item If $Z$ is any other topological space, and $g\colon Z\to X$ is a function for which for every $x\in X$ the function $h_x\colon Z\to\bbR$ defined by $h_x(z)=d(x,g(z))$ is continuous, then $g$ itself is continuous.
\end{enumerate}
\par\noindent{\small {\bf Tip. } ``There exists a unique'' means two things: ``there exists'', and ``if/once exists, it is unique''. Both require a proof!}
\vfill\noindent{\bf Problem 5. } Let $(X_n,d_n)$ be a sequence of metric spaces whose diameters are at most $1$: $\forall n\in\bbN,\, \forall x,y\in X_n,\,d_n(x,y)\leq 1$. Prove that the product $X=\prod_nX_n$ is metrizable.
\vfill\par\noindent{\small {\bf Tip. } Once you have finished writing an exam, if you have time left, it is always a good idea to go back and re-read and improve everything you have written, and maybe even completely rewrite any parts that came out messy.}
\vfill
\centerline{\bf Good Luck!}
\vfill
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