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\begin{center}\textbf{Problem Set 5 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 4/16/24} \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 Let $X$ be a topological space.  If $X$ is a $T_1$-space, then $X$ is a $T_0$-space.

\item Let $X$ be a topological space.  If $X$ is a $T_2$-space, then $X$ is a $T_1$-space.

\item Let $X$ and $Y$ be nonempty topological spaces.  The product space $X \times Y$ is a $T_0$-space iff both $X$ and $Y$ are $T_0$-spaces.

\item Let $X$ and $Y$ be nonempty topological spaces.  The product space $X \times Y$ is a $T_1$-space iff both $X$ and $Y$ are $T_1$-spaces.



\end{enumerate}


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