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\begin{center}\textbf{Problem Set 5 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 3/9/20} \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.  Five of these problems will be selected (by Jon) for grading, with each worth 4 points.

\begin{enumerate}


\item {[Baker 4.2.7]} Prove Theorem 4.2.1.

\item {[Baker 4.2.8]} Prove Theorem 4.2.3.

\item {[Baker 5.3.14]} Prove that the set $\bm{\times} \{F_i:i \in \Lambda \}$ given in example 4.3.15(d) is a closed subset of the product space $X$.

\item Suppose that the indexing set $\Lambda$ in \#3 were changed to be (0,1).  Is $\bm{\times} \{F_i:i \in \Lambda \}$ still closed in the product space $X$?

\item {[Baker Th 5.1.10]} Two nonempty subsets $A$ and $B$ of a topologocal space $X$ are said to be $separated$ provided that $\mathrm{Cl}(A) \cap B = A \cap \mathrm{Cl}(B) = \varnothing$.  Prove that a space $X$ is connected iff $X$ is not the union of two separated sets.

\item {[Baker 5.2.7]} Complete the proof of Theorem 5.2.1.

\item {[Baker 5.2.11]} Prove Corollary 5.2.12.

\item {[Baker 5.2.12]} Prove Corollary 5.2.13

\end{enumerate}


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