\documentclass{homework651}
\include{hwextras}
\usepackage{graphicx}
\usepackage[all,cmtip]{xy}
\def\net<#1>{\left<#1\right>}

\newcommand{\bbB}{\mathbb{B}}
\doclabel{Math F651: Homework 10}
\docdate{Due: April 14, 2023}
\begin{document}
\begin{problems}

\problem The point of this exercise is to settle some details from the 
proof of the Brower fixed point theorem.  We suppose $f:\bbB\to \bbB$ is
continuous and that $f$ does not have a fixed point.
\begin{subproblems}
\item Prove that for all $x\in \bbB^2$ there exists a unique
$t(x)\in[1,\infty)$ such that $f(x)+t(x)(x-f(x))\in S^1$.
\item Define
\[
r(x) = f(x)+t(x)(x-f(x)),
\]
so $r:\bbB\to S^1$.  The graph of $r$ is a subset of $\bbB\times S^1$.  
We wish to show that $r$ is continuous, and since $S^1$ is compact and Hausdorff
it is enough to show that the graph of $r$ is closed.  Do so.  Hint:
Suppose $(x_n,r(x_n))\to (x,z)\in \bbB\times S^1$.  Now show that $z=r(x)$.
We'll discuss in the problem session what a boon the closed graph theorem is here.
\end{subproblems}

\problem Crossley 6.6

\problem 4-23 (a,b)

\problem Lee 8-5

\problem Lee 7-2 (Start after Monday)

\problem Lee 7-3 (ditto)

\end{problems}

\end{document}

