\documentstyle{article}
\begin{document}
\title{Topology Ph.D. Qualifying Exam}
\author{K. Lesh \and G. Martin}
\date{January 16, 1999}
\maketitle
\noindent
This exam has been checked carefully for errors. If you find what
you believe to be an error in a question, report this to the proctor.
If the proctor's interpretation still seems unsatisfactory to you, you may
alter the question so that in your view it is correctly stated, but
not in such a way that it becomes trivial.
\bigskip
\section*{Section 1}
Do 3 of the following 5 problems.
\begin{enumerate}
\item Let $X$ be a topological space and let $\cal{F}$ denote a family
of subsets of $X$. Consider the set
$$W = \{ \cal{F} | \mbox{$\cal{F}$ has the finite intersection property}\}.$$
Partially order $W$ by inclusion. Show that
\begin{enumerate}
\item $W$ has a maximal element.
\item If ${\cal{H}} \in W$ is a maximal element and if $V \cap H \neq
\emptyset$ for all $H \in \cal{H}$, then $V \in H$.
\end{enumerate}
\item Prove that none of the spaces $[0,1]$, $[0,1)$, and $(0,1)$ is
homeomorphic to any of the others.
\item If $Y$ is compact and $Z$ is Hausdorff, show that the
projection map $\pi: Y \times Z \rightarrow Z$ is a closed map.
\item Call a topological space $X$ {\underline{totally disconnected}}
if the only connected subsets of $X$ are one-point sets.
\begin{enumerate}
\item Show that any product of totally disconnected sets is totally
disconnected (in the product topology).
\item Consider a countably infinite product of discrete two-point
spaces, {\em i.e.,}\/ $\Pi^{\infty}_{i=1} \{0,1\}$, again using the
product topology. Prove that this product is not discrete.
Note, however, that by (a), this space is totally disconnected!
\end{enumerate}
\item A family of sets $\cal{F}$ is said to be {\underline{locally
finite}} if for any point $x \in X$ and any open neighborhood. $U$ of
$x$, there are at most a finite number of elements of ${\cal{F}}$ that
intersect $U$ non-trivially.
\begin{enumerate}
\item Show that if ${\cal F}$ is a locally finite family of closed
sets then $\bigcup_{F \in {\cal{F}}} F$ is closed.
\item Show that if $\cal{F}$ is a locally finite closed cover of $X$
and $f: X \rightarrow Y$ has the property that for $F \in {\cal{F}}$,
$f|_{F}$ is continuous, then $f$ is continuous.
\end{enumerate}
\end{enumerate}
\newpage
\section*{Section 2}
Do 3 of the following 5 problems.
\begin{enumerate}
\item Let
$$T^2 = {[0,1]\times[0,1]\over\{(x,0)\sim(x,1)\}\cup
\{(0,y)\sim(1,y)\}}$$
be the 2-torus and consider the quotient space $X =
T^2/\{[(x,0)]\}$. The space $X$ is known as the pinched torus.
As a surface it can be represented as sketched below. Find $\pi_1(X)$.
\bigskip
\bigskip
\bigskip
\bigskip
\item Let $P^2$ be real projective space. The space $P^2$ can be
obtained from the disk $D^2$ by identifying $x \sim -x$ if $\|x\| =
1$. If $p_1, p_2 \in \hbox{int}D^2$, show that $P^2 -\{[p_1], [p_2]\}$
is homotopy equivalent to the figure eight, $S^1\vee S^1$, and find the
fundamental group of $P^2 -\{[p_1], [p_2]\}$.
\item Suppose that $F : X\times I \to Y$ is a homotopy
such that there is $y_0\in Y$ $F|X\times \partial I = y_0$. Define
$F^{-1}(x,t) = F(x,1-t)$. Show the $F*F^{-1}$ is homotopic to the
constant map $y_0$. Here $*$ denotes concatenation and is defined
when $F|X\times 1 = G|X\times 0$ by the expression
\[F*G(x,t) = \left\{\begin{array}{ll}F(x,2t) & 0\le t \le {1\over 2} \\
G(x,2t-1) & {1\over 2} \le t \le 1
\end{array}\right.\]
\item Let $A \subseteq X$ be a subspace of $X$ and let
$i: A \rightarrow X$ be the natural inclusion map. We call
A a {\underline{retract}} of $X$ if there exists a continuous
map $r : X \rightarrow A$ such that $r\circ i$ is the identity
map of $A$.
Suppose that $A \subseteq X$ is a retract and that $\pi_{1} A$ is
normal in $\pi_{1}X$. Show that
$ \pi_{1}X \cong \pi_{1}A \times (\pi_{1} X/ \pi_{1}A)$.
\item Let $p\colon X \to Y$ be a covering projection and let
$\gamma \colon I \to Y$ be a path. Fix a point $x_0$ in
$p^{-1}(\gamma(0))$. Show that there exists a
unique path $\tilde\gamma \colon I \to X$ with the property that
$\tilde\gamma(0) = x_0$ and $p\circ \tilde\gamma = \gamma$.
\end{enumerate}
\end{document}