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\begin{center}\textbf{Problem Set 4 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 3/7/22} \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.  Ten of these problems will be selected (by Jon) for grading, with each worth 2 points.

\begin{enumerate}

\item {[Baker 3.2.2]}

\item {[Baker Ch 3 R12]} If a function $f:\mathbb{R} \to \mathbb{R}$ is $\mathscr{C-U}$ continuous, then $f$ is $\mathscr{U-U}$ continuous.

\item {[Baker Ch 3 R13]} Any two discrete topological spaces are homeomorphic.

\item {[Baker Ch 3 R14]} Any one-to-one, onto function between two discrete topological spaces is a homeomorphism.

\item {[Baker Ch 3 R17]} If $(X,\mathscr{T})$ and $(Y,\mathscr{S})$ are homeomorphic topological spaces, then any one-to-one function from $X$ onto $Y$ is a homeomorphism.

\item Prove Theorem 10 in \S 4.1:  Let $(X,\mathscr{T})$ and $(Y,\mathscr{S})$ be topological spaces.  If $A$ and $B$ are closed subsets of $X$ and $Y$, respectively, then $A \times B$ is a closed subset of $X \times Y$.

\item Prove Lemma 5.1.10: Let $U$ be a nonempty subset of $\mathbb{R}$ that is bounded below and let $n$ be the greatest lower bound for $U$.  If $I$ is any open interval containing $m$, then $I \cap U \neq \varnothing$.

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

\item Prove that topological spaces $(X,\mathscr{T})$ and $(Y,\mathscr{S})$ are connected iff $X \times Y$ with the product topology is connected.

\item Prove Theorem 5.3.4.

\end{enumerate}


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