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\begin{center}\textbf{Problem Set 8 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 4/23/18} \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 {[Baker Review 8.2]} If $(X, \mathscr{T})$ is any topological space, then there is a metric $d$ such that the metric topology induced by $d$ is the same as $\mathscr{T}$.

\item {[Baker Review 8.5]} If there are two different metrics for a given set, then the corresponding metric topologies are different.

\item {[Baker Review 8.7]} If $(X,d)$ is any nonempty metric space, then for any positive integer $N$ there exist distinct points $x$ and $y$ in $X$ such that $d(x,y)<1/N$.

\item {[Baker Review 8.10]} If $X$ is a discrete topological space, then $X$ is metrizable.

\item {[Baker Review 8.12]} Every compact metric space is complete.

\item {Baker Review 8.13]} Every complete metric space is compact.

\end{enumerate}


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