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\begin{center}\textbf{Problem Set 6 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 4/8/20} \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 {[Baker 6.1.7]} Let $\mathbb{R}$ have the usual topology.  Show (directly from the definition) that an open interval $(a,b)$ is not compact.

\item {[Baker 6.1.18]} Give an example of a topology $\mathscr{T}$ on $\mathbb{R}$ for which there is a closed and bounded subset $A$ of $\mathbb{R}$ that is not compact.

\item {[Baker 6.1.18]} Give an example of a topology $\mathscr{S}$ on $\mathbb{R}$ for which there is a compact subset $A$ of $\mathbb{R}$ that is neither closed nor bounded.

\item {[Baker 6.2.5]} Let $X$ be a compact topological space and let $Y$ be a Hausdorff topological space.  Show that if $f:X \to Y$ is a continuous function, then the inverse image of each compact subset of $Y$ is a compact subset of $X$.

\item {[Baker 6.2.8]} Prove that compactness is a topological property.
\end{enumerate}


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