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\begin{center}\textbf{Problem Set 5 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 4/22/22} \end{center}

\noindent You are expected to do the following problems to a high standard (i.e., at least well enough to be published in a textbook) for full credit.  Four of these problems will be selected (by Jon) for grading, with each worth 5 points.

\begin{enumerate}
 
\item Prove Theorem 8.1.9: Let $(X,d)$ be a metric space and let $U \subseteq X$.  Then $U$ is open with respect to the metric topology iff for each $x \in U$, there exists $r>0$ such that $B_r(x) \subseteq U$. 
 
\item {[Baker 8.R.1]} If $X$ is any set, then there is a metric $d$ for $X$.

\item {[Baker 8.R.10]} If $X$ is a discrete topological space, then $X$ is metrizable.
 
\item {[Baker 8.R.11]} If $X$ is an indiscrete topological space, then $X$ is metrizable.

\end{enumerate}


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