Math Analysis Homework - Sem 2 Week 5

T1

2. 设 $E \subset \mathbb{R}^2, P_0$ 为 $E$ 的聚点, 证明存在 $\mathbb{R}^2$ 中的点列 $\{P_n\} \subset E$ ( $P_n \neq P_0, n \in \mathbb{N}$ ), 使得 $\lim_{n \to \infty} P_n = P_0$ .

$P_0$ 是 $E$ 的聚点,即对 $P_0$ 的任意开球邻域 $U=B(P_0,r)$ , $E\cap U\ne \varnothing$ .

则取 $P_1\in B(P_0,1)\cap E$ ,取 $P_n\in B(P_0,\dfrac12 \|P_{n-1},P_0\|)\cap E$ ,则 $\|P_{n-1}-P_0\|\to 0,P_{n-1}\to P_0$ .

T2

3. 对于下列 $\mathbb{R}^2$ 中的点集 $E$ , 求 $E$ 的内核 $E^o$ , 边界集 $\partial E$ , 导集 $E'$ , 并判断哪些是开集、闭集、有界集、开区域和闭区域.

(1) $E = \{(x, y) \mid 0 < x^2 + y^2 < 1\}$ ;
(3) $E = \{(x, y) \mid x^2 + y^2 \leqslant 1\} \cup \{(x, y) \mid y = 0, 1 \leqslant x \leqslant 2\}$ ;
(5) $E = \{(x, y) \mid (x^2 + y^2)(y^2 - x^2 + 1) \leqslant 0\}$ ;
(7) $E = \left\{(x, y) \mid x = \frac{1}{n}, y = \frac{1}{m}; m, n \in \mathbb{N}\right\}$ ;

(1):

\begin{gathered} \operatorname{Int}E=E \\ \partial E=\{ (x,y)|x^2+y^2\in \{ 0,1 \} \} \\ E'=\{ (x,y)|x^2+y^2\in [0,1] \} \end{gathered}

是开集,有界集,开区域,不是闭集,闭区域

(2):

\begin{gathered} \operatorname{Int}E=\{ (x,y)|x^2+y^2<1 \} \\ \partial E=\{ (x,y)|x^2+y^2=1\lor (y=0\land x\in (1,2]) \} \\ E'=E \end{gathered}

是闭集,有界集,不是闭集,不是开区域或闭区域(按照闭区域是开区域闭包的定义)

(3):

\begin{gathered} E=\{ (x,y)|y^2-x^2+1\le 0 \} \cup \{ (0,0) \} \\ \operatorname{Int}E=\{ (x,y)|y^2-x^2+1<0 \} \\ \partial E={ (x,y)|y^2-x^2+1=0 } \cup \{ (0,0) \} \\ E'=E \end{gathered}

是闭集,不是开集,有界集,闭区域或开区域

(4):

\begin{gathered} \operatorname{Int}E=\varnothing \\ \partial E=E \\ E'=\{ (0,0) \} \end{gathered}

是闭集,不是开集,是有界集,不是开区域或闭区域

T3

(1) 设 $E_1, E_2$ 均为闭集, 证明 $E_1 \cup E_2$ 也是闭集. 若 $E_k (k \in \mathbb{N})$ 均为闭集, $\bigcup_{k=1}^\infty E_k$ 是否必为闭集?

(2) 设 $E_1, E_2$ 均为开集, 证明 $E_1 \cap E_2$ 也是开集. 若 $E_k (k \in \mathbb{N})$ 均为开集, $\bigcap_{k=1}^\infty E_k$ 是否必为开集?

(1):

\begin{gathered} \forall x\in (E_1\cup E_2)',\exists \{ x_n \} ,x_n\in E_1\cup E_2,\lim_{n \to \infty} x_n=x, \\ \text{we have } x_n\in E_1\lor x_n\in E_2, \\ \text{let } y_0=0,y_n=\argmin_{k>y_{n-1}} \{ x_k\in E_1 \} \\ \text{ so } x_{y_n}\in E_1,\lim_{n \to \infty} x_n=\lim_{n \to \infty} x_{y_n}\in E_1'\subset E_1\subset E_1\cup E_2 \\ \text{so } (E_1\cup E_2)'\subset (E_1\cup E_2),(E_1\cup E_2)\text{ is closed} \end{gathered}

否, $\cup [\dfrac1n,1]=(0,1]$ 不是闭集.

(2):

\begin{gathered} x\in (E_1\cap E_2) \implies x\in E_1,x\in E_2 \\ \implies \exists x\in B(x,r_1)\subset E_1,x\in B(x,r_2)\subset E_2 \\ \implies x\in B(x,\min (r_1,r_2))\subset (E_1\cap E_2) \\ \implies E_1\cap E_2 \text{ is open} \end{gathered}

T4

5. 设 $E \subset \mathbb{R}^2$ , 证明 $E$ 的内核 $E^o$ 必为开集, $E$ 的导集 $E'$ 必为闭集.

(1):

\begin{gathered} \forall x\in \operatorname{Int}E \\ \exists r>0 \ s.t.\ x\in B(x,r)\subset E \\ \text{then } \forall y\in B(x,\dfrac r3),y\in B(y,\dfrac r3)\subset B(x,r)\subset \operatorname{Int}E \\ \text{so } x\in B(x,\dfrac r3)\subset E,\operatorname{Int}E \text{ is open} \end{gathered}

(2):

\begin{gathered} \forall x\in E'',\exists \{ x_n \},x_n\in E' ,\lim_{n \to \infty} x_n=x \\ \exists \{ y_{n,m} \},y_{n,m}\in E ,\lim_{n \to \infty} y_{n,m}=x_m \\ \text{construct } z_n=y_{k,n}, \text{where } \forall i>k-1,|y_{n,m}-x_m| <\dfrac1{2^n} \\ \text{then } z_n\in E,\lim_{n \to \infty} z_n=\lim_{n \to \infty} x_n=x\in E' \\ \implies E''\subset E',E' \text{ is open} \end{gathered}