due Wednesday, November 3rd by 10pm (Waterloo local time)
In order to receive full credits, you must show all the work leading to your solution. You may of
course work together with your classmates, but you must write up the solutions independently.

Problems
$# 1$ Let $E \subseteq \mathbb{R}^{d}$ be a non-empty compact set. The following questions are independent.
(a) (4 pts) Prove that for all $\varepsilon>0$ there exist $N \in \mathbb{N}$ and $x_{1}, \cdots, x_{N} \in E$ such that $E \subseteq \cup_{j=1}^{N} B_{\varepsilon}\left(x_{j}\right)$
(b) (4 pts) Suppose $\left{F_{i}\right}_{i \in I}$ is an arbitrary collection of closed subsets of $\mathbb{R}^{d}$ such that $F_{i} \subseteq E$ for all $i \in I$ and that $F_{i_{1}} \cap \cdots \cap F_{i_{N}} \neq \emptyset$ for any $N \in \mathbb{N}$ and $i_{1}, \cdots, i_{N} \in I$. Prove that $\cap_{i \in I} F_{i} \neq \emptyset$

#2 Prove that the following sets are compact.
(a) $(5 \mathrm{pts}) E={0} \cup\left(\cup_{n=1}^{\infty} \overline{B_{\frac{1}{n}}\left(a_{n}\right)}\right) \subseteq \mathbb{R}^{d}$, where $\left(a_{n}\right){\mathbb{N}}$ is a given sequence in $\mathbb{R}^{d}$ converging to 0 . (b) $(7$ pts $) E={0} \cup\left{\frac{1}{n} \mid n \in \mathbb{N}\right} \cup\left{\frac{1}{n}+\frac{1}{m} \mid n, m \in \mathbb{N}\right} \subseteq \mathbb{R}$.

$# 3$ (10 pts) Let $E, K$ be disjoint subsets of $\mathbb{R}^{d}$, with $E$ closed and $K$ compact. Prove that there exists $\alpha>0$ such that $|x-y| \geq \alpha$ for all $x \in E$ and $y \in K .$ (First prove that for all $y \in K$ there exists $r{y}>0$ such that $B_{r_{y}}(y) \cap E=\emptyset$.)

$# 4$ (8 pts) Let $E$ be an uncountable subset of $\mathbb{R}^{d}$. Prove that there exists $x \in E$ such that $B_{\delta}(x) \cap E$ is uncountable for all $\delta>0$.

### 5 (12 pts) Let $E$ be a subset of $\mathbb{R}^{d}$. Prove that the following two statements are equivalent.

(i) $E$ is compact.
(ii) For any infinite subset $F \subseteq E$, there exists $x \in E$ such that $B_{\delta}(x) \cap F$ is infinite for all $\delta>0$