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\begin{center}\textbf{Problem Set 3 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 2/15/16} \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 Does the set $\{(a,+\infty)|a \in \mathbb{R} \}$ form a topology on $\mathbb{R}$?
\item {[Baker 2.2.12]} Determine if the function $f:\mathbb{R} \to \mathbb{R}$ given by
    $$f(x)=\left\{  \begin{array}{r c}
  2 & \mathrm{if }\, x \ge 1 \\ 
  -2 & \mathrm{if }\, x<1
 \end{array} \right.$$
is  \begin{enumerate}
    \item $\mathscr{U-U}$ continuous
    \item $\mathscr{U-H}$ continuous
    \item $\mathscr{U-C}$ continuous
    \item $\mathscr{H-U}$ continuous
    \item $\mathscr{H-H}$ continuous
    \item $\mathscr{C-H}$ continuous
    \item $\mathscr{C-C}$ continuous.
\end{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 that $V-U$ is open.
\item {[Baker 2.3.14]} Let $A$ and $B$ be subsets of a topological space $(X, \mathscr{T})$.  Show that $(X-\mathrm{Cl}(A)) \cup (X-\mathrm{Cl}(B)) \subseteq X-\mathrm{Cl}(A \cap B)$.  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
$$X-\mathrm{Cl}(A\cup B)=(X-\mathrm{Cl}(A))\cap(X-\mathrm{Cl}(B)).$$
\item {[Baker 2.4.10]} Show that if $A$ is a subset of a topological space, then Int(Int($A$)) = Int($A$).
\end{enumerate}

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