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\begin{center}\textbf{Problem Set 7 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 4/4/16} \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.

\begin{enumerate}

\item {[Baker 5.R.21]} If $\mathbb{R}$ has the usual topology an $f$ is a one-to-one continuous function from $\mathbb{R}$ onto $\mathbb{R}$, then $f$ is a homeomorphism.

\item {[Baker 6.1.16]} Prove the Bolzano-Weierstrass Theorem (Theorem 6.1.22).

\item {[Baker 6.1.17]} For both the topologies $\mathscr{H}$ and $\mathscr{C}$, describe the bounded infinite subsets of $\mathbb{R}$ which do not have limit points.

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

\item {[Baker 6.1.19]} Prove that if $(X,\mathscr{T})$ is a compact topological space and $\mathscr{S}$ is any topology on $X$ that is coarser than $\mathscr{T}$, then $(X,\mathscr{S})$ is compact.

\item {[Baker 6.1.20]} Give an example to show that a set $X$ can have topologies $\mathscr{T}$ and $\mathscr{F}$ with $\mathscr{F}$ finer than $\mathscr{T}$,  $(X,\mathscr{T})$ compact, and  $(X,\mathscr{F})$ not compact.

\item {[Baker 6.2.11]} Complete the proof of Theorem 6.2.16.

\item {[Baker 6.2.12]} Prove Theorem 6.2.19.

\item {[Baker 7.1.12]} Show that the product of $T_0$ spaces is $T_0$.

\item {[Baker 7.1.12]} Let $X$ be a $T_1$ space with $A \subseteq X$.  Prove that if $x$ is a limit point of $A$ and $U$ is an open set containing $x$, the $U \cap A$ is an infinite set.

\item {[Baker 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 $X \times Y$.

\end{enumerate}


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