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\begin{center}\textbf{Problem Set 2 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 2/4/22} \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 {[Baker 2.3.13]} Let $U$ be a closed set and let $V$ be an open set in a topological space.  Show that $U-V$ is closed and $V-U$ is open.

\item {[Baker 2.3.14]} Let $A$ and $B$ be subsets of a topological space $(X,\mathscr{T})$.  Show that $$(X- \mathrm{Cl}(A)) \cup (X-\mathrm{Cl}(B)) \subseteq X-\mathrm{Cl}(A \cap B).$$

\item {[Baker 2.3.14]} Let $A$ and $B$ be subsets of a topological space $(X,\mathscr{T})$.  Show that $$(X- \mathrm{Cl}(A)) \cup (X-\mathrm{Cl}(B)) \neq X-\mathrm{Cl}(A \cap B).$$

\item {[Baker 2.3.15]} Let $A$ and $B$ be subsets of a topological space $(X,\mathscr{T})$.  Show that $$X-\mathrm{Cl}(A \cup B) = (X-\mathrm{Cl}(A)) \cap (X-\mathrm{Cl}(B)).$$

\end{enumerate}


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