%In this file enter, in the order you wish them to appear
%in your exam, each exam problem. The \problem command sequence
%has three parameters, thus \problem{#1}{#2}{#3},
%where #3 is the problem itself, and must be a single paragraph
%(so don't have any blank lines in it; you can use \vskip0in
%to get the effect of a new paragraph). Parameter #2 should
%be a number; it specifies how many inches of vertical space 
%should be skipped following the problem (thereby leaving
%space for the students to get their answer). Parameter #1
%is the point value of the problem.

%There is also another command that can be used: \comment{#1}{#2},
%which inserts whatever you enter for parameter #2, and skips
%a space in inches specified by #1. Use the comment command to insert
%a short stretch of material that is not a problem, for example
%a section header, to divide the exam up into parts. Note that 
%\comment behaves exactly the same as \problem except that it involves no
%problem number nor a point value. In particular, if your comment will not 
%fit in the space remaining on the page, then the whole comment is
%put on the next page.

\comment{.2}{Section I}

\problem{20}{.2}{Let $X$ be a topological space, and let
$A$ be a subset of $X$. Let ${\cal T}_A'$ be the collection of all 
open subsets $U$ of $X$ such that either $U$ is empty or $U$ meets
$A$. Let ${\cal T}_A''$ be the collection of all 
open subsets $U$ of $X$ such that either $U$ is empty or $U$ contains
$A$. \vskip0in Give a proof or a counterexample for each of the following
two statements:
\itemitem{(a)} The collection ${\cal T}_A'$ is a topology on $X$.
\itemitem{(b)} The collection ${\cal T}_A''$ is a topology on $X$.}

\comment{.2}{Section II}

\problem{20}{.2}{Give a proof or a counterexample for each of the following
two statements:
\itemitem{(a)} Let $X$ and $Y$ be metric spaces with finitely many
points. Then any bijective map from $X$ to $Y$ is a homeomorphism.
\itemitem{(b)} Any bijective, continuous map of topological 
spaces is a homeomorphism.}

\comment{.2}{Section III}

\problem{20}{.2}{Give an example of a {\scaleby{1440} big}
category and then solve {\scaleby{\magstep3} $x3z=\Gamma$}.}

\problem{20}{.2}{Find the determinant of the following matrix:
$$A=\left(\matrix{x & 1\cr
x & 1\cr
x & 1\cr
x & 1\cr
x & 1\cr
x & 1\cr
x & 1\cr
x & 1\cr
x & 1\cr
x & 1\cr
x & 1\cr
x & 1\cr
x & 1\cr
x & 1\cr
x & 1\cr}\right)$$}