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\begin{center}\textbf{Problem Set 1 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 2/5/24} \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 2.1.5]} Prove that if $F$ is any finite subset of $\mathbb{R}$. then $\mathbb{R} - F$ is an open set.

\item {[Baker 2.1.9]} Use Definition 2.1.6 to show that $g(x) =  \begin{cases} -3 \; \mathrm{if} \, x<1 \\ \enspace \: 3 \, \mathrm{if} \;  x \ge 1 \\ \end{cases}$ is not a continuous function.

\item {[Baker 2.1.10]} Complete the proof of Theorem 2.1.8.

\item {[Baker2.2.7]} Show that $\mathscr{U}$ is finer than the finite complement topology for $\mathbb{R}$.


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

\item {[Baker 2.4.14]} Let $A$ be a subset of a topological space.  Prove that $\mathrm{Cl}(A) = \mathrm{Int}(a) \cup \mathrm{Bd}(A)$.

\item {[Baker 2.5.10]} Let $(X,\mathscr{T})$ be a topological space, $\mathscr{B}$ a base for $\mathscr{T}$, and $A \subseteq X$.  Show that $x \in \mathrm{Cl}(A)$ iff for each $B \in \mathscr{B}$ with $x \in B$, $B \cap A \neq \varnothing$.

\item In a space $(X,\mathscr{T})$ any collection of open sets whose union equals $X$ and that is closed under finite intersection is a base for $\mathscr{T}$.
\end{enumerate}


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