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\begin{center}\textbf{Problem Set 6 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 4/6/18} \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 7.1.5.2]} The $T_1$-property is a topological property.

\item {[Baker 7.1.8.1]} If $X$ is a $T_1$-space and $A \subseteq X$, then $A$ is a $T_1$ space.

\item {[Baker 7.1.9]} Let $X$ and $Y$ be topological spaces.  Prove that if $Y$ is a $T_2$-space and $f:X \to Y$ is a continuous one-to-one function, then $X$ is a $T_2$-space.

\item {[Baker 7.1.10]} 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.

\item {[Baker 7.1.15]} Prove that if $(X,\mathscr{T})$ is a finite $T_1$-space, then $\mathscr{T}$ is the discrete topology.

\item {[Baker 7.1.16]} Prove that if $X$ is a $T_1$-space and $A \subseteq X$, then the set of limit points of $A$, $A'$ is a closed subset of $X$.

\end{enumerate}


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