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\noindent
\parbox{1.95in}{\bf Math 4710/5710}
\hfill {\Large \bf Homework \#8} \hfill
\parbox{1.95in}{\bf \hfill Due: Wed., Apr. 2${}^\mathbf{nd}$\!, 2025}

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%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

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\noindent
Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force

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\begin{description}
\item[\#1 Totally Disconnected:] We say a topological space is {\it totally disconnected} if the only connected subspaces are singleton subsets and the empty set. 
\begin{enumerate}[(a)]
\item Carefully explain why each of the following is totally disconnected: \quad\\
(i) Any space with the discrete topology \quad (ii) $\mathbb{R}_\ell$ (reals with the lower limit topology) \quad (iii) $\mathbb{Q}$ 
\item Let $X$ be given the finite complement topology.\\ \force\hfill Explain why $X$ is totally disconnected when finite but connected when infinite.
\end{enumerate}

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\item[\#2 Connecting with Quotients:] Let $p:X \to Y$ be a quotient map.\\ \force\hfill Show $X$ is connected implies $Y$ is connected.

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{\it Note:} The converse is false. Consider $X=(0,1] \,\dot\cup\, [2,3)$. Glue the two intervals together so $1 \sim 2$ to get $Y$. Then $Y = X/\!\sim$ is homeomorphic to an open interval. Thus the projection map $\pi:X \to Y$ in this case sends a disconnected space to a connected one.

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\item[\#3 Following the Right Path:] Let $X$ and $Y$ be topological spaces.
\begin{enumerate}[(a)]
\item Suppose $X$ and $Y$ are path connected. Show $X \times Y$ is path connected.

{\it Note:} This result is true for arbitrary products. 

\item Let $X$ be path connected and $f:X \to Y$ continuous. Show $f(X)$ is path connected.

\item  Let $X_j$ (for $j \in J$) be path connected spaces such that $\bigcap_{j\in J} X_j \not= \emptyset$. Show $\bigcup_{j \in J} X_j$ is path connected.

\item {\bf [Grad.]} We had a theorem saying if $A \subseteq B \subseteq \overline{A}$ and $A$ is connected, then $B$ is connected. Is this result true if ``connected'' is replaced by ``path connected''? 
\end{enumerate}

\item[\#4 Open About Being Connected:] {\bf [Grad.]} Let $U$ be an open and connected subset of $\mathbb{R}^2$.\\ \force\hfill Show $U$ is path connected.

{\it Suggestion:} Consider $u_0 \in U$ and the set $V = \{ u \in U \;|\; u_0 \text{ and } u \text{ are connected by a path in } U\}$.\\ \force\hfill Show $V$ is clopen.  
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