\documentclass{homework651}
\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: 10:00pm March 7, 2025}

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

\begin{problems}
\problem Suppose $q: X\to Y$ is a quotient map and that each fiber of $q$ is connected.
Show that if $Y$ is connected, then so is $X$.

\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}

\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 Suppose the spaces $X_\alpha$, $\alpha \in I$ are all connected
and nonempty, and let $a$ be a point in $X=\prod_{\alpha \in I} X_\alpha$.
\begin{subproblems}
\item Given any finite set $K\subset I$, let $X_K$ denote the subspace
of $X$ where $x_\alpha = a_\alpha$ for all $\alpha \not \in K$.  Show that 
each $X_K$ is connected.
\item Show that $Y = \cup_{K} X_K$ is connected.
\item Show that $\overline Y = X$ and conclude that $X$ is connected.
\end{subproblems}

\begin{wrapfigure}{r}{1.0in}
    \vskip-0.5cm
    \includegraphics[height=1.8in]{Suspension.pdf}
    \end{wrapfigure}
    \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 $\Sigma S^n$.
    
\problem Lee Problem 4-4

\problem Lee Problem 4-5

\problem Lee Problem 4-11

\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 You are not permitted to use any form of AI to assist you in any part of this exam.
\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 5 will be a hints session.
\end{itemize}

\end{problems}
\end{document}
