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\lhead{\large \textbf{Math 461}}
\chead{\Large \textbf{Homework 7}}
\rhead{\large \textbf{DUE 10/29/2023}}
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{\Large\underline{\textbf{Exercises:}}}

\S $17, 18$

\begin{enumerate}

\item Prove each of the following topological spaces is Hausdorff.
	\begin{enumerate}
	\item A set $X$ with an order relation $ < $ and the order topology.
	
	\item A product $X\times Y$ with the product topology where $X$ and $Y$ are Hausdorff spaces.
	
	\item A subspace $Y\subset X$ with the subspace topology where $X$ is a Hausdorff space.	
	\end{enumerate}

\item Let $X$ be a topological space. Show that $X$ is Hausdorff if and only if the \textbf{diagonal}
		\[
			\Delta:=\{ (x,x)\mid x\in X\}
		\]
is a closed subset of $X\times X$ with the product topology.

\item Consider the collection $\mathcal{T}=\{U\subset \mathbb{R} \mid \mathbb{R}\setminus U\text{ is finite}\}\cup\{\emptyset\}$.
	\begin{enumerate}
	\item Show that $\mathcal{T}$ is a topology on $\mathbb{R}$. We call this the \textbf{finite complement topology}.
	
	\item Show that the finite complement topology is $T_1$: given distinct points $x,y\in\mathbb{R}$ there exists open sets $U$ and $V$ with $x\in U\not\ni y$ and $x\not\in V\ni y$.
	
	\item Show that the finite complement topology is not Hausdorff.
	
	\item Find all the points that the sequence $(\frac1n)_{n\in\mathbb{N}}$ converges to in the finite complement topology.
	\end{enumerate}

\item Let $X$ be a set with two topologies $\mathcal{T}$ and $\mathcal{T}'$ and let $i\colon X\to X$ be the identity function: $i(x)=x$ for all $x\in X$. Equip the domain copy of $X$ with the topology $\mathcal{T}$ and the range copy of $X$ with the topology $\mathcal{T}'$.
	\begin{enumerate}
	\item Show that $i$ is continuous if and only if $\mathcal{T}$ is finer than $\mathcal{T}'$.
	
	\item Show that $i$ is a homeomorphism if and only if $\mathcal{T}=\mathcal{T}'$.
	
	\end{enumerate}

\item Consider the functions $f,g\colon \mathbb{R}^2 \to \mathbb{R}$ defined by
	\[
		f(x,y)=x+y \qquad \text{ and }\qquad g(x,y)=x-y.
	\]
	\begin{enumerate}
	\item Show that if $\mathbb{R}$ and $\mathbb{R}^2$ are given the standard topologies, then $f$ and $g$ are continuous.
	
	\item Suppose $\mathbb{R}$ is given the lower limit topology and $\mathbb{R}^2=\mathbb{R}\times\mathbb{R}$ is given the corresponding product topology. Determine and prove the continuity or discontinuity of $f$ and $g$.
	\end{enumerate}


\item[6*.] In this exercise you will establish a homeomorphism between the following two subspaces of $\mathbb{R}^2$:
	\[
		X:= \mathbb{R}^2\setminus\{(0,0)\} \quad \text{ and }\qquad Y:=\{(x,y)\in\mathbb{R}^2\mid x^2 + y^2 >1\}.
	\]
Throughout, $\mathbb{R}^2$ will have the standard topology and $X$ and $Y$ will have their subspace topologies.	
	\begin{enumerate}
	\item Define a function $\|\cdot\|\colon \mathbb{R}^2 \to [0,+\infty)$ by $\| (x,y)\| =(x^2+y^2)^{1/2}$. Show that this function is continuous when $[0,+\infty)\subset \R$ is given the subspace topology. [\textbf{Hint:} think geometrically.]
	
	
	\item Show that $X=\{(x,y)\in\mathbb{R}^2\mid \|(x,y)\|>0\}$ and $Y =\{(x,y)\in \mathbb{R}^2\mid \|(x,y)\| >1\}$.
	
	\item Show that $f\colon X\to \mathbb{R}^2$ defined by $f(x,y) = \frac{1}{\|(x,y)\|}(x,y)$ is continuous.
	
	\item Find continuous functions $g\colon X\to Y$ and $h\colon Y\to X$ satisfying $g\circ h(x,y)=(x,y)$ and $h\circ g(x,y)=(x,y)$, and deduce that $X$ and $Y$ are homeomorphic.
	
	\end{enumerate}
$^{*}$ Challenge Problem!

\end{enumerate}


	
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