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UF MTG 4302/5316\\
Introduction to Topology 1\\
Fall 2023
\begin{center}
{\large \bf Homework 4}\\
Due Friday, October 20, anytime, on Canvas
\end{center}

{\bf Reading.}
Munkres \S21 -- \S25


{\bf Problems.}
\begin{itemize}

\item \S18 \#9(a,b).
Let $\{A_\alpha\}$ be a collection of subsets of $X$; let $X=\cup_\alpha A_\alpha$.
Let $f\colon X \to Y$; suppose that $f|A_\alpha$ is continuous for each $\alpha$.

(a) Show that if the collection $\{A_\alpha\}$ is finite and each set $A_\alpha$ is closed, then $f$ is continuous.

(b) Find an example where the collection $\{A_\alpha\}$ is countable and each $A_\alpha$ is closed, but $f$ is not continuous.

\item \S19 \#6.
Let $\textbf{x}_1, \textbf{x}_2, \ldots$ be a sequence of points of the product space $\prod X_\alpha$.
Show that this sequence converges to the point $\textbf{x}$ if and only if the sequence $\pi_\alpha(\textbf{x}_1), \pi_\alpha(\textbf{x}_2), \ldots$ converges to $\pi_\alpha(\textbf{x})$ for each $\alpha$.
Is this fact true if one uses the box topology instead of the product topology?

\item \S21 \#3.
Let $X_n$ be a metric space with metric $d_n$, for $n\in \Z_+$.

(a) Show that
$\rho(x,y)=\max\{d_1(x_1,y_1),\ldots,d_n(x_n,y_n)\}$ 
is a metric for the product space $X_1\times \ldots \times X_n$.

(b) Let $\overline{d_i}=\min\{d_i,1\}$.
Show that
$D(x,y) = \sup\left\{\tfrac{\overline{d_i}(x_i,y_i)}{i}\right\}$
is a metric for the product space $\prod X_i$.

\emph{Remark: For (b), compare the proof of Theorem~20.5, which is the specific case when $X_n=\R$ and $d_n$ is the standard metric on $\R$, for all $n$.}

\item \S22 \#2(a).
Let $p\colon X\to Y$ be a continuous map.
Show that if there is a continuous map $f\colon Y\to X$ such that $p\circ f$ equals the identity map of $Y$, then $p$ is a quotient map.


\end{itemize}

\vfill

{\bf Recommend Problems (not to turn in).}
\begin{itemize}

\item \S17 \#13.

\item \S19 \#1.

\item \S20 \#4.

\end{itemize}

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