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\noindent K. Lesh \hfill{23 Sept. 1995}\\
\noindent G. Thompson\\
\begin{center}
Topology Qualifying Exam
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The Ph.D. qualifying exam committee tries to proofread the examinations as
carefully as possible. Nevertheless, the exam may contain misprints. If
you are convinced a problem has been stated incorrectly, mention this to the
proctor and indicate your interpretation in your solution. In such cases do not
interpret the problem in such a way that it becomes trivial.\\
\noindent Directions: Do four problems in each section. Budget your
time. Write your solution for each question on a separate page. \\
\noindent Section I
\begin{enumerate}
\item Let $A$ and $B$ be closed subspaces of a topological space $X$
with $X = A \bigcup B$. Suppose that $f: A \rightarrow Y$ and
$g: B \rightarrow Y$ are continuous, and $f(x)=g(x)$
for all $x \in A \bigcap B$.
\noindent Prove that $h: X \rightarrow Y$ by
$$ h(x) = \left\{ \begin{array}{ccc}
f(x) & \mbox { if } & x \in A \\
g(x) & \mbox { if } & x \in B
\end{array} \right. $$
is continuous. Is it necessary for both $A$ and $B$ to be closed?
Discuss.
\item Let $f : X \rightarrow Y$ by a quotient map. Let $Y$ be connected
and suppose that for each $y \in Y$, $f^{-1}(y)$ is connected. Prove
that $X$ is connected.
\item Let $I$ be a non empty index set, let
$\{ X_{\alpha} | \alpha \in I \}$ be a family of topological spaces, and let
$A_{\alpha} \subseteq X_{\alpha}$ for each $\alpha$.
\begin{enumerate}
\item Show that if $A_{\alpha}$ is closed in $X_{\alpha}$ for
each $\alpha$, then $\prod A_{\alpha}$ is closed in
$\prod X_{\alpha}$.
\item Show that ${\overline{\prod A_{\alpha}}}
= \prod {\overline {A_{\alpha}}}$.
\item Prove or disprove: If $A_{\alpha}$ is open in $X_{\alpha}$
for each $\alpha$, then $\prod A_{\alpha}$ is open in
$\prod X_{\alpha}$.
\end{enumerate}
\item Let $D$ be the closed unit disk in the complex plane. Let
$\sim$ be the equivalence relation on $D$ defined by
$z_{1} \sim z_{2}$ if and only if $z_{1} = z_{2}$ or
$|z_{1}| = |z_{2}| < 1$. Is the quotient
topological space Hausdorff? (Prove your assertion.)
\item State the definition of compactness for topological spaces. Prove
{\underline{from}} {\underline{your definition}}
that the closed unit interval $[0,1]$ is compact.
\end{enumerate}
\newpage
\noindent Section II
\begin{enumerate}
\item Define what it means for $Y$ to be a strong deformation retract of
$X$, where $Y \subseteq X$ are topological spaces. Prove that if
$i : Y \rightarrow X$ is the inclusion map and $y \in Y$, then
the induced homomorphism
$i_{*}: \pi_{1} (Y,y) \rightarrow \pi_{1} (X,y)$
is an isomorphism.
\item Prove by any method you know that:
\begin{enumerate}
\item $\Re $ is not homeomorphic to $\Re^{2}$.
\item $\Re^{2}$ is not homeomorphic to $\Re^{3}$.
\end{enumerate}
\item Let $X_{1}$ and $X_{2}$ be two copies of $S^{2}$ and let $N_{1},
S_{1}$ and $N_{2}, S_{2}$ be the north and south poles of $X_{1}$ and
$X_{2}$, respectively. Define $X$ to be the quotient space obtained by
identifying $N_{1}$ with $N_{2}$ and $S_{1}$ with $S_{2}$. Compute
the fundamental group of $X$ by using the Seifert-van Kampen theorem.
\item \begin{enumerate}
\item Define a covering space.
\item State the main theorem about path lifting and covering
spaces.
\item Let $S^{1} \bigvee \Re P^{2}$ be the one point union of the
circle and two dimensional real projective space, {\it i.e.}
the quotient space obtained by taking the disjoint union
of $S^{2}$ and $\Re P^{2}$ and then identifying a single point
$x \in S^{2}$ with a single point in $ y \in \Re P^{2}$.
Describe the universal cover of $S^{1} \bigvee \Re P^{2}$.
\item Describe the fundamental group of $S^{1} \bigvee \Re P^{2}$.
\end{enumerate}
\item Prove that $\Re^{2}$ cannot be retracted to $S^{1}$.
\end{enumerate}
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