Problem Set 2

(a) Prove Theorem A: Let $(X, d)$ be a nonempty complete metric space. Let $\Omega_1, \Omega_2, \ldots \subseteq X$ be open dense sets. Then $\bigcap_{n \geq 1} \Omega_n$ is dense in $X$.

(b) A set $P$ is perfect iff it is closed with no isolated points.

(b1) Show $P$ is perfect iff $P = P'$ (the derived set).

(b2) Prove Theorem B: Let $(X, d)$ be a complete metric space and $P$ a nonempty perfect subset. Then $P$ is uncountable.

Let $(X, \mathcal{T})$ be a topological space, $f_n : X \to \mathbb{R}$ continuous for all $n \geq 1$, with $f_n(x) \to f(x)$ pointwise. Let $D$ be the discontinuity set of $f$. Show $D$ is meagre. Follow the strategy using $P_n(\varepsilon) = \{x : |f(x) - f_n(x)| \leq \varepsilon\}$, $G(\varepsilon) = \bigcup_n \text{int}(P_n(\varepsilon))$, $Y = \bigcap_k G(2^{-k})$, $F_n(\varepsilon) = \{x : |f_n(x) - f_m(x)| \leq \varepsilon \,\forall m \geq n\}$, $L(\varepsilon) = \bigcup_n \text{int}(F_n(\varepsilon))$.

Let $\lambda = \sum_{n=1}^\infty 10^{-n!}$.

(a) Show $\lambda \notin \mathbb{Q}$.

(b) Show $\bigcap_{m \geq 1} \bigcap_{n \geq 1} \bigcup_{p/q \in \mathbb{Q}} (p/q - 1/(nq^m), p/q + 1/(nq^m)) = \bigcap_{k \geq 1} \bigcup_{p/q \in \mathbb{Q}, q > 1} (p/q - 1/q^k, p/q + 1/q^k)$.

(c) Let $q_k = 10^{k!}$, $p_k = \sum_{\ell=1}^k 10^{k! - \ell!}$. Show $|\lambda - p_k/q_k| < 1/q_k^k$.

Let $(X, \mathcal{T})$ be topological, $p \in X$, $C(p) = \bigcup_{S \ni p, S \text{ connected}} S$. Show $C(p)$ is connected.

For every $a \in \mathbb{R}$ and $k \geq 1$, there exists $x \in (a + \pi/k, a + 3\pi/k)$ with $|\cos(ka) - \cos(kx)| \geq 1$.

Let $f(x) = \sum_{k=0}^\infty 5^{-k} \cos(200^k x)$.

(a) Show $f$ is continuous.

(b)–(i) Show $f$ is nowhere differentiable by constructing $x_n \to a$ with $|f(x_n) - f(a)|/|x_n - a| \to \infty$.