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UF MTG 4302/5316\\
Introduction to Topology 1\\
Fall 2023
\begin{center}
{\large \bf Homework 6}\\
Due Monday, November 20, anytime, on Canvas
\end{center}

{\bf Reading.}
Munkres \S32 -- \S33, \S43.


{\bf Problems.}
\begin{enumerate}

\item Define what it means for a topological space to be regular.

Show that every compact Hausdorff space is regular.

\emph{Remark: It is furthermore true that every compact Hausdorff space is normal (Theorem~32.3), which suffices since every normal space is regular. Read Theorem~32.3, but don't use it. Also don't use Lemma~26.4 --- instead, I'm asking you to use the proof technique of Lemma~26.4 to reprove a variant thereof.}

\emph{Remark: This problem has appeared on the first year exam before; see for example (\footnote{\url{https://gma.math.ufl.edu/wp-content/uploads/sites/130/FY-Topology-1-2023_08.pdf}}).}

\item Assume that $f\colon X\to Y$ is continuous and surjective.

(a) If $X$ is Lindel\"{o}f, show that $Y$ is also.

(b) If $X$ is separable, show that $Y$ is also.

\emph{Remark: This problem has appeared on the first year exam before; see for example (\footnote{\url{https://gma.math.ufl.edu/wp-content/uploads/sites/130/FY-Topology-1-2022_08.pdf}}).}




\end{enumerate}

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

\item \S30 \#4.

\item \S32 \#1.

\end{itemize}

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