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UF MTG 4302/5316\\
Introduction to Topology 1\\
Fall 2023
\begin{center}
{\large \bf Homework 7}\\
Due Monday, December 4, anytime, on Canvas
\end{center}

{\bf Reading.}
Munkres \S43--44, \S48, \S50.


{\bf Problems.}
\begin{enumerate}

\item
Prove that every compact Hausdorff space is normal. You may follow the proof of Theorem~32.3 in the book (which states this result).

\item \S43 \#4(forward direction).
Show that if the metric space $(X,d)$ is complete, then for every nested sequence $A_1\supset A_2\supset\ldots$ of nonempty closed sets of $X$ such that $\diam A_n \to 0$, the intersection of the sets $A_n$ is nonempty.

\item \S44 \#1.
Given $n$, show there is a continuous surjective map $g\colon I \to I^n$.
(\emph{Hint:} Consider $f\times f \colon I\times I\to I^2 \times I^2$, where $f\colon I\to I^2$ is the surjective map from Theorem~44.1.)

\item \S48 \#4.
Show that if every point $x$ of $X$ has a neighborhood that is a Baire space, then $X$ is a Baire space.
(\emph{Hint:} Use the open set formulation of the Baire condition.)

\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_05.pdf}}).}




\end{enumerate}

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

\item \S32 \#2.

\item \S43 \#5.

\item \S44 \#2.

\item \S48 \#11.

\end{itemize}

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