\documentstyle[12pt,amssymb]{article}
\voffset -1in
\hoffset -1in
\textheight 9in
\textwidth 7in
\begin{document}
\large
\pagestyle{empty}
.
\vspace{1.5in}
\begin{center} TOPOLOGY QUALIFYING EXAMINATION \\
\bigskip
Time: \ \ Three hours. \\
\bigskip
Do FOUR problems in each section. \end{center}
\vspace{5in} \hfill April 1997
\newpage
\noindent {\bf Section A}\\
\begin{enumerate}
\item Let $A_1, A_2, \ldots, A_k$ be a collection of connected subsets of a topological space $X$. Suppose that each $A_i$ is {\it not} disjoint from all the remaining $A_i$'s. Show that $Y=A_1 \cup A_2 \ldots \cup A_k$ is connected.
%2
\item Let ${\Bbb R}$ have the ``finite-closed'' topology i.e., the closed sets other than ${\Bbb R}$ and $\emptyset$ are the finite subsets of ${\Bbb R}$. Is this space:
\begin{enumerate}
\item connected?
\item Hausdorff?
\item metrizable?
\item Also describe the compact subspaces of this space.\\
\noindent Justify your answer in each case.
\end{enumerate}
%3
\item \begin{enumerate}
\item Let $X$ be Hausdorff topological space. Show that $X$ is normal iff each neighborhood of a closed set $F$ contains the closure of some neighborhood of $F$.
\item Let $A$ be an indexing set and let $X$ be a topological space. Define the disjoint union of subsets $X_\alpha$ of $X$ to be $\cup (\alpha \times X_\alpha)$ where $\alpha \in A$ and $A \times X$ is topologized by taking a basis of open sets to be $\alpha \times U_\alpha$ where $U_\alpha$ is open in $X_\alpha$. Suppose that the $X_\alpha$ are closed in $\cup X_\alpha (\subset X)$ and $A$ is finite. Show that $j : \cup (\alpha \times X_\alpha) \longrightarrow \displaystyle \bigcup_{\alpha \in A} X_\alpha$ is an identification map.
\end{enumerate}
%4
\item Let $p : X \longrightarrow Y$ be a continuous map of Hausdorff spaces. Suppose that for any $y \in Y, p^{-1}(y)$ is compact and second countable and there exists an open neighborhood $U_y$ such that $p^{-1}(U_y)$ is homeomorphic to $p^{-1}(y) \times U_y$. Prove that $p$ is a closed map.
%5
\item Let $f : X \longrightarrow Y$ be a continuous map of topological spaces. Define the {\it graph} of $f$ to be the subset $\{(x, f(x) : x \in X\}$ of $X \times Y$. Prove that:\
\begin{enumerate}
\item if $Y$ is Hausdorff then the graph of $f$ is closed.
\item if $X$ is connected then the graph of $f$ is connected.
\end{enumerate}
\end{enumerate}
\newpage
\noindent {\bf Section B}
\bigskip
\begin{enumerate}
\item.
\vspace{1in}\\
\begin{enumerate}
\item Compute the fundamental group of the space obtained by identifying the points $x$ and $y$ pictured above on the {\it solid} three hole torus.
\item Compute the fundamental group of the space obtained by identifying the points $x$ and $y$ on the {\it surface} of the three hole torus.
\end{enumerate}
%2
\item \begin{enumerate}
\item Define the join (or product) of two compatible loops $\alpha, \beta$ in a topological space $X$. If $\gamma$ is a third loop find an explicit homotopy between the loops $(\alpha \cdot \beta) \cdot \gamma$ and $\alpha \cdot (\beta \cdot \gamma)$.
\item The space $G$ is a topological group meaning that $G$ is a group and also a Hausdorff topological space such that the multiplication and map taking each element to its inverse are continuous operations. Given two loops based at the identity $e$ in $G$, say $\alpha (s)$ and $\beta (s)$, we have two ways to combine them: $\alpha \cdot \beta$ (join of loops as in (a)) and secondly $\alpha \beta$ using the group multiplication. Show, however, that these constructions give homotopic loops.
\end{enumerate}
%3
\item .%
\vspace{1.75in}\\
\noindent The annulus shown has its edges identified to form a surface $S$. [``Annulus'' means we start with the shaded region \hspace{1.5in} i.e., the interior of the inner rectangle is ignored].\\
Identify $S$ as one of the standard closed surfaces in the following two ways:
\begin{enumerate}
\item By cutting along $PQ$, represent $S$ as a polygon with edges identified in pairs.
\item By cutting around the circle $\gamma$.
\end{enumerate}
%4
\item Let $S^1$ denote the unit circle with center the origin in ${\Bbb R}^2$.
\begin{enumerate}
\item Define $f : S^1 \longrightarrow S^1$ by $f(x)=-x$. Prove that $f$ is homotopic to the identity map.
\item Let $g : S^1 \longrightarrow S^1$ be a map which is not homotopic to the identity. Show that $g(x)=-x$ for some $x$ in $S^1$.
\end{enumerate}
%5
\item Let $\pi : \tilde{X} \longrightarrow X$ be a covering space.
\begin{enumerate}
\item Show that $\pi$ is an open map.
\item Show that if $\tilde{X}$ is homeomorphic to $X \times \pi^{-1}(p)$, where $\pi(q)=p$ with $q,p$ being base points in $\tilde{X}$ and $X$, respectively, there is a continuous map $s : X \longrightarrow \tilde{X}$ such that $\pi \circ s =$ identity on $X$.
%c
\item Assume now that $X$ is path-connected. Show that if there exists a continuous map $s : X \longrightarrow \tilde{X}$ such that $\pi \circ s=$ identity on $X$, then $\tilde{X}$ is disconnected unless $\pi^{-1}(p)$ consists precisely of $q$.
\end{enumerate}
\end{enumerate}
\end{document}