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\begin{center}\textbf{Problem Set 5 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 3/26/18} \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 5.3.4]} Complete the proof of Lemma 5.3.1 by showing that the function $f$ is continuous.

\item Find an open cover of $(a,b)$ in $(\mathbb{R},\mathscr{U})$ which does not have a finite subcover.

\item {[Baker 6.1.10]} Let $(X, \mathscr{T})$ be a topological space and let $A \subseteq X$.  The set $A$ is compact iff every cover of $A$ consisting of $\mathscr{T}$-open subsets of $X$ has a finite subcover.

\item {[Baker 6.1.16]} Every bounded infinite subset of $(\mathbb{R}, \mathscr{U})$ has at least one limit point.

\end{enumerate}


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