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\noindent
\parbox{1.95in}{\bf Math 4710/5710}
\hfill {\Large \bf Homework \#9} \hfill
\parbox{1.95in}{\bf \hfill Due: Fri., Apr. 11${}^\mathbf{th}$\!, 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 Finitely Many Problems:] Let $Y_1,\dots,Y_n$ be compact subspaces of some topological space $X$. 
\begin{enumerate}[(a)]
\item Show that $Y_1 \cup \cdots \cup Y_n$ is compact. [A finite union of compact subspaces is still compact.]
\item Give a counterexample to show that this can fail when we union infinitely many compact subspaces.
\end{enumerate}

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\item[\#2 Take it to the Limit:] Suppose $x_1,x_2,\dots$ is a sequence in a topological space $X$ such that $x_n \to x$\\ \force\hfill (i.e., the sequence converges to $x \in X$). Prove $C = \{x\} \cup \{x_1,x_2,\dots\}$ is compact.

\vspace{0.15in}

\item[\#3 You'll be O$K$:] Recall $K=\{1,1/2,1/3,\dots\}$ and $\mathcal{B}_K = \{ (a,b), (a,b)-K \;|\; a,b \in \mathbb{R} \text{ and } a<b\}$ is a basis\\ \force\hfill for $\mathbb{R}_K$ ($\mathbb{R}$ with the $K$-topology).

It is helpful to note that if $0 \not\in U$, then $U$ is open in $\mathbb{R}_K$ if and only if $U$ is open in $\mathbb{R}$ with the standard topology. Only open neighborhoods of $0$ cause problems for us!

\begin{enumerate}[(a)]
\item Show $I=[0,1]=\{ x \in \mathbb{R} \;|\; 0 \leq x \leq 1\}$ is not compact when viewed as a subspace of $\mathbb{R}_K$.

{\it Note:} By our note above, since $K$-open and standard open sets coincide when $0$ is not present, any closed interval (e.g., $[1,2]$ or $[-10,-3]$) avoiding $0$ would be compact!

\item Show $\mathbb{R}_K$ is connected. 

\item {\bf [Grad.]} Show $\mathbb{R}_K$ is not path connected.
\end{enumerate}

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