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\noindent
\parbox{1.95in}{\bf Math 4710/5710}
\hfill {\Large \bf Homework \#6} \hfill
\parbox{1.95in}{\bf \hfill Due: Wed., Mar. 5${}^\mathbf{th}$\!, 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 Getting Closure:] Let $A$, $B$, and $C_i$ (where $i \in I$) be subsets of a topological space $X$. Show\dots
\begin{enumerate}[(a)]
\item If $A\subseteq B$, then $\overline{A} \subseteq \overline{B}$
\item $\overline{A\cup B} = \overline{A} \cup \overline{B}$
\item $\displaystyle \bigcup_{i\in I} \overline{C_i} \subseteq \overline{\bigcup_{i\in I} C_i}$
\item Give an example of the containment in part (c) being proper (i.e., not equal).
\end{enumerate}

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\item[\#2 Open to Inequality (Shame on you Hausdorff!):] Let $X$ be a Hausdorff (i.e., $T_2$) space.\\ \force\hfill Show the diagonal $\Delta = \{ (x,x) \;|\; x \in X\}$ is closed in $X \times X$ (with the product topology).

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\item[\#3 Weird Closure:] {\bf [Grad.]} Recall $\mathbb{R}_\ell$ is the real numbers equipped with the lower limit topology. Let $\mathbb{R}_{\text{rat.}-\ell}$ be the real numbers equipped with the topology generated by the basis $\mathcal{C}=\{ [a,b) \;|\; a,b \in \mathbb{Q}\}$. We could call this the rational lower limit topology. {\it Note:} We previously showed that the topology on $\mathbb{R}_\ell$ is strictly finer than that of $\mathbb{R}_{\text{rat.}-\ell}$. 

Determine the closures of $A=(0,\sqrt{2})$ and $B=(\sqrt{2},4)$ in the (a) standard topology, (b) lower limit topology, and (c) rational lower limit topology.

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\item[\#4 Continuous Products:] Let $f:A \to B$ and $g:C \to D$ be continuous functions between topological spaces. Consider $f \times g:A \times C \to B \times D$ defined by $(f \times g)(a,c)=(f(a),g(c))$. Prove $f \times g$ is continuous. 

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