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\begin{center}\textbf{Problem Set 2 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 1/29/20} \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.  Four of these problems will be selected (by Jon) for grading, with each worth 5 points.

\begin{enumerate}

\item {[Baker 2.1.7]} For $i=1,2,...,n$ let $I_n=(a_i,b_i)$ be an open interval. Show that $\bigcap\{I_i:i=1,2,...,n\}$ is either the empty set or an open interval.\\

\item {[Baker 2.1.8]} Use Definition 2.1.6 to show that $f(x)=\begin{cases}\enspace -3&\mathrm{if } \, x < 1\\
\enspace \enspace 3&\mathrm{if } \, x \geq 1\end{cases}$ is not a continuous function.\\

\item {[Baker 2.1.9]} Use Definition 2.1.6 to show that $f(x)=\begin{cases}\enspace 1/x&\mathrm{if } \, x \neq 0\\
\enspace \enspace 0&\mathrm{if } \, x = 0\end{cases}$ is not a continuous function.\\

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

\item {[Baker 2.2.10]} Show that the collection $\mathscr{C}$ given in Example 2.2.3 is a topology for $\mathbb{R}$.\\

\item {[Baker 2.2.12]} Determine if the function $f:\mathbb{R} \to \mathbb{R}$ given by
\[f(x)=\begin{cases}\enspace \, 2&\mathrm{if } \, x > 1\\
-2&\mathrm{if } \, x \leq 1\end{cases}\]
is \begin{itemize} 
  \item[(a)] $\mathscr{U-U}$ continuous
  \item[(b)] $\mathscr{U-H}$ continuous
  \item[(c)] $\mathscr{U-C}$ continuous
  \item[(d)] $\mathscr{H-U}$ continuous
  \item[(e)] $\mathscr{H-H}$ continuous
  \item[(f)] $\mathscr{C-H}$ continuous
  \item[(g)] $\mathscr{C-C}$ continuous
  \end{itemize}
\vspace{12pt}

\item {[Baker 2.3.14]} Let $A$ and $B$ be subsets of a topolpogical space $(X,\mathscr{T})$.  Show that $(X-\mathrm{Cl}(A))\cup(X-\mathrm{Cl}(B)) \subseteq X-\mathrm{Cl}(A \cap B)$.  Find an example that shows these sets are not in general equal.\\

\item {[Baker 2.3.15]} Let $A$ and $B$ be subsets of a topological space $(X,\mathscr{T}).$ Show that 
\[X-\mathrm{Cl}(A \cup B) = (X - \mathrm{Cl}(A)) \cap(X-\mathrm{Cl}(B)).\]
\vspace{6pt}

\end{enumerate}


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