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\begin{center} M.A. TOPOLOGY EXAM \hspace{.5in} SPRING 1998 \end{center}
\begin{center} L. Bentley and G. Thompson \end{center}
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\begin{quote}
\noindent Do no more than five (5) questions. If you think there is a misprint in a question state your query clearly and try to interpret it in a non-trivial manner.
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The Examination Committee tries to proofread the exams 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.
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\noindent {\bf Exam is \underline{two} hours. Do five (5) problems.}
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\begin{enumerate}
\item Define what it means for a topological space to be {\it disconnected}. Prove that a space is disconnected if and only if there is a continuous map from the space onto the discrete two point space \{0,1\}.
\item State whether the following propositions are true or false. If they are true prove them. If they are false give a counterexample.
\begin{enumerate}
\item In a compact topological space a closed subspace is compact.
\item In a metric space a compact subspace is closed.
\item In a Hausdorff space a compact subspace is closed.
\end{enumerate}
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\item Prove that a topological space $X$ is Hausdorff if and only if the diagonal $\Delta = \{(x,x) \in X \times X : x \in X\}$ is closed in $X\times X$ where $X\times X$ has the product topology induced by $X$.
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\item Prove that a continuous bijection from a compact topological space onto a Hausdorff topological space is necessarily a homeomorphism.
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\item Define the product topology $X\times Y$ on topological spaces $X,Y$. Prove from your definition that each of the projections from $X\times Y$ \, to \, $X$ and $Y$, respectively, are both continuous and open maps.
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\item Define what it means for $p : X\rightarrow Y$ to be a quotient map of topological spaces or, equivalently, for $p$ to be an identification map. Verify in detail that $\exp : {\Bbb R} \rightarrow {\Bbb R}^2$ given by $\exp (t) =(\cos t, \sin t)$ gives an identification map of ${\Bbb R}$ onto the unit circle.
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\item Prove that as subspaces of ${\Bbb R}$ with the usual topology the open interval (0,1) is not homeomorphic to the half-open interval [0,1).
\end{enumerate}
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