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UF MTG 4302/5316\\
Introduction to Topology 1\\
Fall 2023
\begin{center}
{\large \bf Homework 1}\\
Due Wednesday, September 6, in class
\end{center}

{\bf Reading.}
Munkres \S1 -- \S7, \S9


{\bf Problems.}
\begin{itemize}

\item \S5 \#3.
Let $A=A_1\times A_2\times \ldots$ and $B=B_1\times B_2\times \ldots$.

(a) Show that if $B_i \subset A_i$ for all $i$, then $B \subset A$.

(b) Show the converse of (a) holds if $B$ is nonempty.

(c) Show that if $A$ is nonempty, each $A_i$ is nonempty.
Does the converse hold?
\emph{Comment: No need to answer this question about the converse.}

(d) What is the relation between the set $A\cup B$ and the cartesian product of the sets $A_i \cup B_i$?
What is the relation between the set $A\cap B$ and the cartesian product of the sets $A_i \cap B_i$?
\emph{Comment: No need to prove your answer.}

\item \S6 \#3.
Let $X$ be the two-element set $\{0,1\}$.
Find a bijective correspondence between $X^\omega$ and a proper subset of itself.

\item \S6 \#4a.
Let $A$ be a nonempty finite simply ordered set.

(a) Show that $A$ has a largest element.
[\emph{Hint:} Proceed by induction on the cardinality of $A$.]

\item \S7 \#3.
Let $X$ be the two-element set $\{0,1\}$.
Show there is a bijective correspondence between the set $\mathcal{P}(\Z_+)$ and the cartesian product $X^\omega$.

\end{itemize}

\vfill

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

\item \S6 \#2.

\item \S7 \#4.

\item \S7 \#5ef.

\end{itemize}

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