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\begin{center}\textbf{Problem Set 8 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 4/11/16} \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.

\begin{enumerate}

\item {[Baker 7.R.1]} Any indiscrete topological space is not Hausdorff.

\item {[Baker 7.R.2]} Any indiscrete topological space with two or more points is not Hausdorff.

\item {[Baker 7.R.3]} Singleton subsets of $T_2$-spaces are closed.

\item {[Baker 7.R.4]} Singleton subsets of $T_1$-spaces are closed.

\item {[Baker 7.R.5]} Singleton subsets of $T_0$-spaces are closed.

\item {[Baker 7.R.6]} Subspaces of regular spaces are regular.

\item {[Baker 7.R.7]} Subspaces of $T_2$-spaces are $T_2$-spaces.

\item {[Baker 7.R.8]} Closed subsets of normal spaces are normal.

\item {[Baker 7.R.9]} Every normal space is regular.

\item {[Baker 7.R.10]} Every normal space is a $T_1$-space.

\item {[Baker 8.1.8]} 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 8.1.9]} If $d$ is the metric on $\mathbb{R}$ given by $d(x,y)=|x-y|$ for all $x,y \in \mathbb{R}$, then the corresponding metric topology for $\mathbb{R}$ is $\mathscr{U}$.

\end{enumerate}


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