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\begin{center}\textbf{Problem Set 8 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 4/20/20} \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.  Five of these problems will be selected (by Jon) for grading, with each worth 4 points.

\begin{enumerate}

\item {[Baker Th. 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 8.2.5]} Complete the proof of Theorem 8.2.5.

\item {[Baker 8.2.5]} Let $(X,d)$ and $(Y,e)$ be metric spaces.  Prove that a function $f:X \to Y$ is continuous iff, for each $x \in X$ and each $\varepsilon > 0$, there exists a $\delta > 0$ such that $f(B_\delta ^d (x) \subseteq B_\varepsilon ^e (f(x))$.

\item {[Baker 8.2.6]} Complete the proof of Theorem 8.2.13.

\item {[Baker Th. 8.3.12]} Let $X$ be a topological space with $A \subseteq X$ and $x \in X$.  If there is a sequence in $A - \{x\}$ which converges to $x$, then $x$ is a limit point of $A$.
\end{enumerate}


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