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UF MTG 4302/5316\\
Introduction to Topology 1\\
Fall 2023
\begin{center}
{\large \bf Homework 3}\\
Due Wednesday, September 27, anytime, on Canvas
\end{center}

{\bf Reading.}
Munkres \S16 -- \S20


{\bf Problems.}
\begin{itemize}

\item \S13 \#6.
Show that the topologies of $\R_\ell$ and $\R_K$ are not comparable.

\emph{These topologies are defined on page 82, and showing that they are not comparable means finding a set that is open in $\R_\ell$ but not open in $\R_K$, and finding a set that is open in $\R_K$ but not open in $\R_\ell$.}

\item \S16 \#1.
Show that if $Y$ is a subspace of $X$, and $A$ is a subset of $Y$, then the topology $A$ inherits as a subspace of $Y$ is the same as the topology it inherits as a subspace of $X$.

\item \S16 \#6.
Show that the countable collection
\[ \{(a,b)\times(c,d)~|~a<b\text{ and }c<d\text{, and }a,b,c,d\text{ are rational}\} \]
is a basis for $\R^2$.

\emph{The standard topology on $\R$ is the order topology, and the standard topology on $\R^2=\R\times\R$ is the product topology.}

\item \S17 \#2.
Show that if $A$ is closed in $Y$ and $Y$ is closed in $X$, then $A$ is closed in $X$.



\end{itemize}

\vfill

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

\item \S13 \#8.

\item \S16 \#4.

\item \S17 \#3.

\end{itemize}

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