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\begin{center}\textbf{Problem Set 6 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 4/18/24} \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 \, [Baker Exercise 7.1.13] Let $X$ be a $T_1$ space.  Prove that if $A$ is a finite subset of $X$, then $A$ does not have a limit point.

\item  \, [Baker Exercise 7.1.14]  Let $X$ and $Y$ be topological spaces and let $f:X \to Y$ be a function.  The $graph$ of $f$ is the set $G(f) = \{(x,y) \in X \times Y : y = f(x)\}$.  Prove that if $f$ is continuous and $Y$ is a $T_2$-space, then $G(f)$ is a closed subset of the product space $X \times Y$.

\item  \, [Baker Exercise 8.2.9] Let $(X,d)$ be a metric space.  Show that the function $e: X \times X \to \mathbb{R}$ given by $e(x,y) = \min \{1, d(x,y)\}$ is a metric for $X$.

\item  \, [Baker Exercise 8.2.10] Show that the metric topology induced by the metric $e$ given in Exercise 9 is the same as the metric topology induced by $d$.



\end{enumerate}


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