\documentclass[minion]{homework651}
% \usepackage{cmacros}
\include{hwextras}
\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage[all,cmtip]{xy}

\def\net<#1>{\left<#1\right>}
\newcommand\RP{{\mathbb R\mathbb P}}
\newcommand\RPn{\RP^n}
\def\abs#1{\left|#1\right|}

\doclabel{Math F651: Take Home Final}
\docdate{Due: 10:00pm Friday, May 2, 2025}

\begin{document}

{\bf Please see the rules below.}

	
\begin{problems}

\problem Let $C(\Reals)$ denote the set of continuous functions from $\Reals$ to $\Reals$.
Show that $C(\Reals)$ is dense in $\Reals^\Reals$ in the product topology, but not dense in the box topology.
	
\problem The M\"obius band is the quotient of $[0,1]\times \Reals$ where
$(0,y)\sim (1,-y)$.
\begin{subproblems}
\item Show that the M\"obius band is a 2-manifold.
\item Show that the M\"obius band is homotopy equivalent to a circle.
\item No rigor please, just a picture or two: what familiar space is the 1-point compactificaiton of the M\"obius band?
\end{subproblems}

\problem You will recall that showing that $\RP^n$ is Hausdorff was a hassle.  Here's 
a slick technique when the quotient you are looking at arises (as it often does!) from a group action.
\begin{subproblems}
\item Lee 3-22 b
\item Show $\RP^n$ is Hausdorff. (Hurrah Closed Map Theorem!)
\end{subproblems}

\problem  Suppose the Earth is a sphere and that temperature is a continuous function of position on Earth.  Prove that there is a point $p$ on Earth where the temperature is the same as at its antipodal
point $-p$.

\begin{wrapfigure}{r}{1.8in}
	\vskip -.5cm
\includegraphics{hw5f1}
\caption{Gluing a half sphere.}
\end{wrapfigure}
\problem  
\begin{subproblems}
\item Show that the upper half sphere $S^{n,+}$
with antipodal points on $\partial S^{n,+}$ identified is homeomorphic to $\RPn$.
\item Consider $X=\{(xy,yz,zx,x^2,y^2,z^2)\in\Reals^6:x^2+y^2+z^2=1\}$.
Prove that this set is homeomorphic to $\RP^2$.	
\end{subproblems}

\problem Suppose $G$ is a topological group, $A$ is a closed subset,
and $B$ is a compact subset.  Show that $AB$ is closed. Hint: Nets!

\problem Recall that a set $A\subset X$ is a retract of $X$ if there 
is a continuous $f:X\to A$ such that $f(a)=a$ for all $a\in A$.
\begin{subproblems}
\item Show that if $X$ is Hausdorff and $A$ is a retract of $X$ then $A$ is closed.
\item Let $A$ be a two point subset of $\Reals^2$. Show that it is not a retract of $\Reals^2$.
\item Show that the closed ball $\overline{\mathbb{B}^2}$ is a retract of $\Reals^2$.
\item Show that $S^1$ is not a retract of $\Reals^2$.
\end{subproblems}

\problem Consider a map $g:S^1 \rightarrow S^1$.  Using 
degree theory techniques, show that $g$ admits a lift
$\tilde g:S^1\rightarrow \Reals$ if and only if $\deg(g)=0$.

\problem Lee 9-5

\problem Compute the fundamental groups of the following spaces:
\begin{subproblems}
\item The projective plane with two points removed.
\item A torus with two points removed.
\item The union of the unit sphere $S^2$ in $\Reals^3$ with the $z$-axis.
\end{subproblems}

\end{problems}
\bigskip
\hrule
\bigskip
{\bf  Rules and format:}
\begin{itemize}

\item You are welcome to discuss this exam with me (David Maxwell) to ask for hints and so forth.
\item  If you find a suspected typo, please contact me as soon as possible and I will
communicate it to the class if needed.
\item You may not discuss the exam with anyone else (including an AI) until after the due date/time.
\item You are permitted to reference either of the two texts used for the class (Lee or Crossley).
No other resources are permitted.
\item Each problem is weighted equally (10 points each).
\item The due date/time is absolutely firm.
\item We will schedule a hints sessions for this exam.
\end{itemize}
 

\end{document}

