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\begin{center}\textbf{Problem Set 7 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 4/13/20} \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 Consider $\mathbb{R}$ with the topology consisting of intervals of the form $(-a,a)$ for $a \in \mathbb{R}^+$, which we call the Hammar topology.  Is the Hammar topology $T_0$?  $T_1$? $T_2$?.  

\item Consider $\mathbb{R}$ with the $\mathscr{H}$ topology.  Is this space $T_0$?  $T_1$? $T_2$?

\item {[Baker 7.1.5]} Prove that the $T_2$ property is a topological property.

\item Suppose that $X$ is a $T_0$ space.  Give an example to show that $X \times Y$ need not be a $T_0$ space.

\item {[Baker 7.1.8]} Prove that if $X$ is a $T_1$-space and $A\subseteq X$, then $A$ is a $T_1$-space.
\end{enumerate}


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