%%%%%%%%%%%%%%%%%%%%%%%%
%                       	    Preamble                                 %
%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass{report}
\usepackage[left=1in,right=1in,top=1in,bottom=1in]{geometry}
\setlength{\headheight}{23pt}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{mathrsfs}
\usepackage{multirow}
\usepackage{verbatim}
\usepackage{url}
\usepackage{yfonts}
\usepackage{fancyhdr}
\pagestyle{fancy}
\usepackage[colorlinks]{hyperref}
\usepackage{wrapfig}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{shapes,decorations,calc,arrows}

\begin{document}


%%%%%%%%%%%%%%%%%%%%%%%%
%                         	     Header                                    %
%%%%%%%%%%%%%%%%%%%%%%%%
\fancyhead{}
\fancyfoot{}
\lhead{\large \textbf{Math 461}}
\chead{\Large \textbf{Homework 8}}
\rhead{\large \textbf{DUE 11/5/2023}}
\cfoot{\thepage}


%%%%%%%%%%%%%%%%%%%%%%%%
%                       Custom Commands                           %
%%%%%%%%%%%%%%%%%%%%%%%%

%\renewcommand{\theenumii}{\roman{enumii}}
%\renewcommand{\labelenumii}{(\theenumii)}
\newcommand{\supp}{\text{supp}}
\newcommand{\Aut}{\text{Aut}}
\newcommand{\Gal}{\text{Gal}}
\newcommand{\Inn}{\text{Inn}}
\newcommand{\Irr}{\text{Irr}}
\newcommand{\Ker}{\text{Ker}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\D}{\mathbb{D}}
\renewcommand{\H}{\mathcal{H}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\K}{\mathcal{K}}
\newcommand{\M}{\mathcal{M}}
\newcommand{\<}{\left\langle}
\renewcommand{\>}{\right\rangle}
\renewcommand{\Re}[1]{\text{Re}\ #1}
\renewcommand{\Im}[1]{\text{Im}\ #1}
\renewcommand{\mod}[1]{(\operatorname{mod}#1)}
\newcommand{\mb}[1]{\mathbb{#1}}
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\ran}[1]{\operatorname{ran}#1}
%\renewcommand{\span}{\text{span}}

\newcommand{\e}{\mathbf{e}}
\renewcommand{\u}{\mathbf{u}}
\renewcommand{\v}{\mathbf{v}}
\newcommand{\w}{\mathbf{w}}
\newcommand{\x}{\mathbf{x}}
\newcommand{\y}{\mathbf{y}}
\newcommand{\0}{\mathbf{0}}


%%%%%%%%%%%%%%%%%%%%%%%%
%                 		    Document                                %
%%%%%%%%%%%%%%%%%%%%%%%%

{\Large\underline{\textbf{Exercises:}}}

\S $18, 19$

\begin{enumerate}

\item Let $A,B,C,D$ be topological spaces and suppose $f\colon A\to B$ and $g\colon C\to D$ are continuous functions. Define a function $f\times g\colon A\times C\to B\times D$ by
	\[
		(f\times g)(a,c) = ((f(a), g(c)).
	\]
Show that $f\times g$ is continuous when $A\times C$ and $B\times D$ are given the product topologies.


\item Let $\mathbb{R}$ and $\mathbb{R}^2$ have their standard topologies.
	\begin{enumerate}
	\item Show that the function $f\colon \mathbb{R}^2\to\mathbb{R}$ defined by $f(x,y)=xy$ is continuous.
	
	\item For each $n\in \mathbb{N}$, show that $p\colon \mathbb{R}\to \mathbb{R}$ defined by $p(x)=x^n$ is continuous.		
		
	\end{enumerate}



\item Let $X$ be a topological space and let $Y$ be a set with an order relation $<$ and the order topology. Suppose $f,g\colon X\to Y$ are continuous.
	\begin{enumerate}
	\item Show that the set $\{x\in X \mid f(x)\leq g(x)\}$ is closed in $X$.
	
	\item Show that the function $h\colon X\to Y$ defined by $h(x):=\min\{f(x),g(x)\}$ is continuous. [\textbf{Hint:} using the pasting lemma.]
	\end{enumerate}
	
	
	
\item Let $\mathbb{R}$ have the standard topology. Consider
	\[
		C = \{ (x_n)_{n\in \mathbb{N}} \in \mathbb{R}^\mathbb{N}\mid x_n\neq 0 \text{ for only finitely many }n\in\mathbb{N}\}.
	\]
That is, $C$ is the set of sequences that are eventually equal to zero.
	\begin{enumerate}
	\item Determine $\overline{C}$ when $\mathbb{R}^\mathbb{N}$ has the box topology.
	
	\item Determine $\overline{C}$ when $\mathbb{R}^\mathbb{N}$ has the product topology.
	\end{enumerate}


	
\item Let $\mathbb{R}$ have the standard topology and consider the functions $f,g\colon \mathbb{R}\to\mathbb{R}$ defined by
	\[
		f(x)= \begin{cases} 1 & x\in \mathbb{Q}\\
							0 & x\in \mathbb{R}\setminus\mathbb{Q} \end{cases},
	\]
and
	\[
		g(x)=\begin{cases} \frac{1}{m} & x\in\mathbb{Q} \text{ with $x=\frac{n}{m}$ for $n\in \mathbb{Z}$ and $m\in \mathbb{N}$ sharing no common factors}\\
							0 & x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}.
	\]
	\begin{enumerate}
	\item Show that $\mathbb{Q}$ and $\mathbb{R}\setminus\mathbb{Q}$ are dense in $\mathbb{R}$.
	
	\item Show that $f$ is \textbf{not} continuous at any $x\in \mathbb{R}$.
	
	\item Show that $g$ is \textbf{not} continuous at any $x\in \mathbb{Q}$.
	
	\item Show that $g$ is continuous at every $x\in \mathbb{R}\setminus\mathbb{Q}$.
	\end{enumerate}



\end{enumerate}


	
\end{document}​