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\begin{center}\begin{large}\textbf{Math 131 - Problem Set 8\\
Due Thursday, Nov 8}\end{large}\end{center}



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\noindent From Munkres: 54.8, 22.2

\begin{enumerate}

\item For $X$ a topological space and $A \subset X$, let $X/A$ denote the quotient space of $X$ where all points in $A \subset X$ are identified to a single point. Prove that $\R^2/A$ is homeomorphic to $\R^2$ when $A=B^2 ( = \{x \mid \|x \| \leq 1\})$, but not when $A=S^1$.

\item Let $X = \R^{n+1} - \{0\}$, $n \geq 0$. In class, we defined an equivalence relation on $X$ by $x \sim y \iff x=\alpha y$ for some $\alpha \in \R$. The quotient space $X / \sim$ is called projective $n$-space, denoted $\R \mathbb{P}^n$. 

\begin{enumerate} 
\item Show that $\R \mathbb{P}^{n}$ is homeomorphic to the quotient space $S^n/ \sim$ where $a \sim b \iff a = -b$.

\item Show that $\R \mathbb{P}^1$ is homeomorphic to $S^1$. 
\end{enumerate}

\item Let $E$, $F$, and $B$ be topological spaces. A \textbf{fiber bundle} with fiber $F$ is a continuous map $f: E \to B$, where every $b \in B$ has an open neighborhood $U \subset B$ satisfying the following:

\begin{itemize}
\item There is a homeomorphism $h: f^{-1}(U) \to U \times F$.
\item If $\pi : U \times F \to U$ is the projection onto the first factor, $f|_{f^{-1}(U)} = \pi \circ h$.
\end{itemize}

\begin{enumerate}
\item Show that a fiber bundle $f: E \to B$ is a quotient map.
\item Show that the M\"{o}bius band (i.e. the quotient of $[0, 1] \times [0, 1]$ by the relation $(0, y) \sim (1, 1 - y))$  is a fiber bundle over $S^1$.
\item Show that if $B$ is connected, $f:E \to B$ is a covering map if and only if $f$ is a fiber bundle with fiber $F$, where $F$ has the discrete topology.

\end{enumerate}



\end{enumerate}

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