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\begin{center}{\large \bf TOPOLOGY QUALIFYING EXAM}\end{center}
\noindent{\bf Do four problems in each section. Explain your arguments clearly and carefully.}\\
\bigskip
{\bf Section 1}
\begin{enumerate}
\item Prove or disprove: if $f : [a,b] \longrightarrow [c,d]$ is continuous, surjective, and increasing then $f$ is a homeomorphism.\\
\item Prove or disprove: in a compact topological space every infinite set has a limit point. \\
%3
\item Let $A$ be a subspace of a topological space $X$. Prove $A$ is disconnected if and only if there exist open subsets $P$ and $Q$ of $X$ such that $A \subset P \cup Q, \ P \cap Q \subset X-A$ \ and \ $P\cap A \neq \emptyset , \ Q \cap A \neq \emptyset.$ \ [{\it Note:} $X - A$ denotes the complement of $A$ in $X$.]\\
\item Prove that if $f : X \longrightarrow Y$ is continuous and surjective, $X$ is compact and $Y$ is Hausdorff then $f$ is an identification map. \\
\item Let $A$ and $B$ be disjoint compact subsets of a Hausdorff space $X$. Then there are disjoint, open sets containing $A$ and $B$. Prove this theorem.
\end{enumerate}
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\noindent {\bf Section 2}\\
\begin{enumerate}
\item A connected topological space $X$ is said to be {\it contractible} if the identity map of $X$ to itself is homotopic to a constant map on $X$. Prove that $X$ is contractible if and only if $X$ has the homotopy type of a point.\\
%2
\item Compute the fundamental groups of the following spaces $X$ by any valid method.
\begin{enumerate}\begin{enumerate}
\item $X$ is a tube with {\it closed} ends and two points are removed.
\item $X$ is a tube with {\it open} ends and two points are removed.
\end{enumerate} \end{enumerate}
\item Prove that the closed unit disk $D$ in ${\Bbb R}^2$ cannot be retracted to the unit circle $S^1$. Deduce that any continuous map $f : D \longrightarrow D$ has a fixed point. [{\it Hint:}\ consider the line joining $x$ to $f(x)$ where $x \in D$.]
\item {Identify the surface whose polygonal symbol is the given octagon. What is the surface's fundamental group?}
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\item \ i. \ Let $f : (0,17) \longrightarrow S^1$ \ by \ $f(x) = e^{2\pi ix}$. Is $f$ a covering map?
\indent \ ii.\ Describe all covering spaces of the Klein bottle and justify that they are \\
\indent \ \ \ \ the only ones.
\end{enumerate}
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