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\lhead{\large \textbf{Math 461}}
\chead{\Large \textbf{Homework 11}}
\rhead{\large \textbf{DUE 12/3/2023}}
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{\Large\underline{\textbf{Exercises:}}}

\S $22$

\begin{enumerate}

\item Let $X$ be a topological space. For a subset $A\subset X$, a \textbf{retraction} of $X$ onto $A$ is a continuous map $r\colon X\to A$ satisfying $r(a)=a$ for all $a\in A$.
	\begin{enumerate}
	\item Let $p\colon X\to Y$ be a continuous map between topological spaces. Show that if there exists a continuous function $f\colon Y\to X$ so that $p(f(y))=y$ for all $y\in Y$, then $p$ is a quotient map.
	
	\item Show that a retraction is a quotient map.
	\end{enumerate}

\item Consider the following subset of $\mathbb{R}^2$:
	\[
		A:=\{(x,y)\in \mathbb{R}^2\mid \text{either }x\geq 0 \text{ or }y=0 \text{ (or both)}\}.
	\]
Define $q\colon A\to \mathbb{R}$ by $q(x,y)=x$. Show that $q$ is a quotient map, but is neither open nor closed.	

\item Let $X$ and $Y$ be topological spaces and let $p\colon X\to Y$ be a surjective map.
	\begin{enumerate}
	\item Show that a subset $A\subset X$ is saturated with respect to $p$ if and only if $X\setminus A$ is saturated with respect to $p$.
	
	\item Show that $p(U)\subset Y$ is open for all saturated open sets $U\subset X$ if and only if $p(A)\subset Y$ is closed for all saturated closed sets $A\subset X$.
	
	\item Show that if $p$ is an injective quotient map, then it is a homeomorphism.
	\end{enumerate}


\item Let $X:=(0,1]\cup [2,3)$, $Y:=(0,2)$, and $Z:=(0,1]\cup (2,3)$ and define maps $p\colon X\to Y$ and $q\colon X\to Z$ by
	\[
		p(t):=\begin{cases} t & \text{if }0<t\leq 1 \\ t-1 & \text{if }2\leq t<3\end{cases} \qquad\text{ and }\qquad q(t):=\begin{cases} t &\text{if } t\neq 2 \\ 1 & \text{otherwise} \end{cases}.
	\]
Equip $X$ and $Y$ with their subspace topologies from $\mathbb{R}$ and equip $Z$ with the quotient topology induced by $q$.
	\begin{enumerate}
	\item Show that $p$ is a quotient map.
	
	\item Show that $q$ is a quotient map.
	
	\item Show that $f\colon Y\to Z$ defined by
		\[
			f(t):=\begin{cases} t & \text{if }0<t\leq 1 \\ t+1 & \text{if } 1<t< 2 \end{cases}
		\]
	is a homeomorphism. [\textbf{Hint:} show $f\circ p = q$.]
	\end{enumerate}

\item Recall that for $\mathbf{x}=(x_1,x_2)\in\mathbb{R}^2$, its norm is $\|\mathbf{x}\|=(x_1^2+x_2^2)^{1/2}$. Consider $X:=\mathbf{R}^2\setminus\{(0,0)\}$ and $S^1:=\{\mathbf{x}\in\mathbb{R}^2\mid \|\mathbf{x}\|=1\}$ equipped with their subspace topologies, where $\mathbb{R}^2$ has its standard topology.
	\begin{enumerate}
	\item Show that $p(\mathbf{x}):=\frac{1}{\|\mathbf{x}\|}\mathbf{x}$ defines a continuous map $p\colon X\to S^1$.
	
	\item Show that $p$ is a quotient map.
	
	\item Define an equivalence relation $\sim$ on $X$ so that the quotient space $X/\sim$ is homeomorphic to $S^1$. Give a geometric description of the equivalence classes.
	\end{enumerate}

\item[6*.] Consider
	\begin{align*}
		X&:=\{\mathbf{x}\in \mathbb{R}^2\mid \|\mathbf{x}\|\leq 1\}\\
		S^2&:=\{\mathbf{x}\in \mathbb{R}^3\mid \|\mathbf{x}\|=1\}.
	\end{align*}
In this exercise you will show a quotient space of $X$ is homeomorphic to $S^2$.
	\begin{enumerate}
	\item Let $S^1:=\{\mathbf{x}\in \mathbb{R}^2\mid \|\mathbf{x}\|=1\}$. Show that $f\colon X\setminus S^1\to \mathbb{R}^2$ defined by
		\[
			f(\mathbf{x}) := \frac{1}{1-\|\mathbf{x}\|}\mathbf{x}
		\]
	is a homeomorphism.
	
	\item Show that $g\colon S^2\setminus\{(0,0,1)\}\to \mathbb{R}^2$ defined by
		\[
			g(\mathbf{x}):= \frac{1}{1-x_3}(x_1,x_2)
		\]
	is a homeomorphism.
	
	\item Show that $p\colon X\to S^2$ defined by
		\[
			p(\mathbf{x}) := \begin{cases} g^{-1}\circ f(\mathbf{x}) & \text{if } \mathbf{x}\in X\setminus S^1 \\ (0,0,1) & \text{otherwise} \end{cases}
		\]
	is a quotient map.
	
	\item Define an equivalence relation on $X$ by $\mathbf{x}\sim\mathbf{y}$ if and only if $p(\mathbf{x})=p(\mathbf{y})$. Describe the quotient space $X/\sim$ and show that it is homeomorphic to $S^2$.
	\end{enumerate}
$^{*}$ Challenge Problem!

\end{enumerate}


	
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