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\noindent
\parbox{2.25in}{\bf Math 4710/5710-101}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Aug. $25^\mathrm{th}$, 2014}

\vspace{0.4in}

\noindent {\large Munkres' Topology \S 2 Problem \#2 (pages 20-21):} Let $f:A \to B$ and let $A_i \subset A$ and $B_i \subset B$ for $i=0$ and $i=1$. Show that $f^{-1}$ preserves inclusions, unions, intersections, and differences of sets:
\begin{enumerate}[(a)]
\item $B_0 \subset B$ \quad $\Longrightarrow$ \quad $f^{-1}(B_0) \subset f^{-1}(B)$.
\item $f^{-1}\left(B_0 \cup B_1\right)  = f^{-1}(B_0) \cup f^{-1}(B_1)$.
\item $f^{-1}\left(B_0 \cap B_1\right)  = f^{-1}(B_0) \cap f^{-1}(B_1)$.
\item $f^{-1}\left(B_0 - B_1\right)  = f^{-1}(B_0) - f^{-1}(B_1)$.

\vspace{0.15in}

\force \hspace{-0.35in} Show that $f$ preserves inclusions and unions only:

\vspace{0.05in}

\item $A_0 \subset A_1$ \quad $\Longrightarrow$ \quad $f(A_0) \subset f(A_1)$.
\item $f(A_0 \cup A_1) = f(A_0) \cup f(A_1)$.
\item $f(A_0 \cap A_1) \subset f(A_0) \cap f(A_1)$; show equality holds if $f$ is injective.
\item $f(A_0 - A_1) \supset f(A_0) - f(A_1)$; show equality holds if $f$ is injective.
\end{enumerate}


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