Mark mentioned that these were good problems in class today and I wanted to post them before I forgot he ever said anything about them.
3.2.1 (a) Where in the proof of $\texttt{Theorem 3.2.3}$ part (ii) does the assumption that the collection of open sets be finite get used?
(b) Give an example of an infinite collection of nested open sets $$O_1 \supseteq O_2 \supseteq \dots$$ whose intersection $\bigcap_{n=1}^{\infty} O_n$ is closed and nonempty.
3.2.2 Let $$ B=\{ \frac{(-1)^nn}{n+1}\mid n=1,2,3,\dots \}.$$
(a) Find the limit points of $B$.
(b) Is $B$ a closed set?
(c) Is $B$ an open set?
(d) Does $B$ contain any isolated points?
(e) Find $\overline{B}$.
3.2.3 Decide whether the following sets are open, closed or neither. If a set is not open, find a point in the set for which there is no $\varepsilon$-neighborhood contained in the set. If a set is not closed, find a limit point that is not contained in the set.
(a) $\mathbb{Q}$
(b) $\mathbb{N}$
(c) $\{x\in\mathbb{R} \mid x>0\}$
(d) $\{ x\in \mathbb{R} \mid 0 < x \leq 1 \} = (0 ,1]$
(e) $\{ 1 + 1/4 + 1/9 + \dots + 1/n^2 \mid n\in\mathbb{N} \}$
3.2.12 Decide whether the following statements are true or false. Provide counterexamples for those that are false, and supply proofs for those that are true.
(a) For any set $A\subseteq \mathbb{R}, \overline{A}^c$ is open.
(b) If a set $A$ has an isolated point, it cannot be an open set.
(c) A set $A$ is closed iff $\overline{A}=A$.
(d) If $A$ is a bounded set, then $s=\sup{A}$ is a limit point of $A$.
(e) Every finite set is closed.
(f) An open set that contains every rational number must be all of $\mathbb{R}$.