\documentclass{homework651}
% \usepackage{cmacros}
\include{hwextras}
\def\ra{\rightarrow}
\def\Reals{\mathbb{R}}
\usepackage{graphicx}
% \usepackage{wrapfig}
\usepackage[all,cmtip]{xy}
\geometry{top=1in, bottom=0.75in, left=1in, right=1in}


\doclabel{Math F651: Take Home Midterm}
\docdate{Due: 9:15am March 10, 2023 }

\begin{document}
See \textbf{Rules} on following page.


\begin{problems}
\problem A subset $A$ of a topological space $X$ is said to be nowhere dense if $\mathrm{Int}\; \overline{A} = \emptyset$.
\begin{subproblems}
\item Let $U$ be an open subset of a topological space. Prove that $\partial U$ is closed and nowhere dense.
\item Let $V$ be a closed and nowhere dense set. Show that $V$ is the boundary of an open set.
\end{subproblems}


\problem Let $f$ and $g$ be continuous maps from a topological space $X$ to a Hausdorff
space $Y$.  Suppose $f=g$ on a dense subset of $X$. Prove that $f=g$. 

\problem {\color{blue} Exercise 4.38}
%Suppose $A$ and $B$ are disjoint compact subsets
%of a Hausdorff space.  Show that there are disjoint open sets $U_A$
%and $U_B$ with $A\subset U_A$ and $B \subset U_B$.

\problem Suppose $X$ and $Y$ are spaces and $Y$ is compact.
Show that the projection $X\times Y \to X$ is a closed map.

\problem Let $G$ be an algebraic group.  We say that $G$ is a {\bf topological group}
if in addition $G$ is a topological space
such that that the multiplication
map $m:G\times G\ra G$ and the inversion map $i:G\ra G$ defined by $m(g,h)=g\cdot h$ and $i(g)=g^{-1}$
are continuous.
\begin{subproblems}
\item
Suppose $G$ is an algebraic group and a topological space.  Show that $G$ is a 
topological group if and only if the map $f:G\times G\ra G$ defined by $f(g,h)=g h^{-1}$
is continuous.
\item Let $G$ be a topological group and let $H$ be a subgroup.  Show that $\overline H$ is
a subgroup.  Hint: that map $f$ from the previous part is continuous. 
% 😉
\end{subproblems}

% \problem Consider the set $\Reals^\omega$. Let $\ell^2$ be the subset of $\Reals^\omega$
% of square summable sequences.  That is, $(x_1,x_2,\ldots)\in \ell^2$ if $x_1^2+x_2^2+\cdots$ is finite.
% We can put a metric on $\ell^2$ by $d(\mathbf{x},\mathbf{y})=\sqrt{(x_1-y_1)^2+\cdots}$ You
% do {\bf NOT} need to verify that this is a metric.  We take the topology on $\ell^2$
% to be the topology generated by this metric.

% Determine if the embedding of $\ell^2\hookrightarrow \Reals^\omega$ is continuous if
% $\Reals^\omega$ is given
% \begin{subproblems}
%   \item the product topology, or
%   \item the box topology.
% \end{subproblems}

\problem Let $\{x_n\}_n$ be a sequence in an arbitrary
product $\prod X_\alpha$. Show that $x_n\to x$ if 
and only if $\pi_\alpha(x_n)\to \pi_\alpha(x)$ for every $\alpha$.
Then show that this result is false if we assume
instead that $\prod X_\alpha$ is given the box topology.

\problem Lee Problem 4-4

\problem Lee Problem 4-5

\problem Lee Problem 4-11

\problem Let $X$ be a topological space.  The {\bf suspension} of $X$, denoted by
$\Sigma X$, is the quotient of $X\times [-1,1]$ where all points of the form $(x,1)$
are identified, and all points of the form $(x,-1)$ are identified.  
Determine, with proof, a familiar space that is homeomorphic to $\Sigma S^n$.

\includegraphics[height=1.5in]{Suspension.pdf}

\newpage
{\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 error or misprint, please contact me as soon as possible and I will
communicate it to the class if needed.
\item You my not discuss the exam with anyone else until after the due date/time.
\item You are permitted to reference Lee or any other topology text you like.  If you use
another text, you must cite it when you use it.
\item You may not consult any internet resources, including search engines.
\item Each problem is weighted equally.
\item The due date/time is absolutely firm.
\item The problem session on March 7 will be a hints session.
\end{itemize}



\end{problems}
\end{document}
