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PMATH 333 Assignment 7代写

$# 1$ 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 $\delta>0$

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