[數學]高微的問題 急

Question

最近做得不太會所以上來問問看1.

(a)Is it true that int(A) ∪int(B) = int(A ∪B)??(b) Is it true that int(A) ∩ int(B) = int(A ∩ B)??(c) Is it true that int(cl(A)) = int(A)? (d) Is it true that cl(A) ∩ A = A?

(e) Is it true that cl(int(A)) = A?

2. (a) Discusswhether the following sets are open or closed:

a. (1, 2) in R

b. [2, 3] in R

c. {r ∈ ]0, 1[| r is rational} i^R d. {(x, y) ∈ R^2 | 0 < x ≤ 1} inR^2

e. {x ∈ R^n | ||x|| = 1} in R^n(b) Determine the interiors, closures, andboundaries of the upper sets.

3.Find the accumulation points of the following sets in R^2:

(a) {(m, n) | m, n integers}

(b) {( p, q) | p, q rational}

(c) {(m/n, 1/n) | m, n integers, n ≠ 0}

(d) {(1/n + 1/m, 0) | n,m integers, n ≠ 0,m ≠ 0}

Answer 1

最佳解答

1.

(a) 錯

反例：令 A, B ⊂ R，A = Q，B = (R＼Q)

than int(A) = φ ，int(B) = φ

int(A)∪int(B) = φ ≠ R = int(A∪B) = int(R)

(b) 對

證明：

(左包含於右)

令 x 屬於 int(A)∩int(B)

則存在 r > 0 使得 D(x , r) 同時包含於 A，且包含於B

=> D(x , r) 包含於 A∩B

=> x 屬於 int(A∩B)

=> int(A)∩int(B) ⊂ int(A∩B)

(右包含於左)

令 x 屬於 int(A∩B)

則存在 r > 0 使得 D(x , r) 包含於 A∩B

=> D(x , r) 同時包含於 A，且包含於B

=> x 屬於 int(A)∩int(B)

=> int(A∩B) ⊂ int(A)∩int(B)

=> int(A∩B) = int(A)∩int(B) 得證

(c) 錯

反例：令 A = Q

int(cl(A)) = int(R) = R，int(A) = φ

(d) 對

證明：由定義，A ⊂ cl(A)

=> cl(A) ∩ A = A

(e) 錯

反例：A 若不是 closed set 則不成立

2.

(a)

a. open

b. closed

c. 非 open 也非 closed

實數線上，任何有理數點的 neighborhood 必然含有無理數點

=> 非 open

而它的補集也因為無理數點的 neighborhood 必然含有有理數點的關係，非open

=> 非 closed

d. 非 open 也非 closed

e. closed

(b)

a. interior : (1 , 2)，closure : [1 , 2]，boundary : {1 , 2}

b. interior : (2 , 3)，closure : [2 , 3]，boundary : {2 , 3}

c. interior : φ，closure : [0 , 1]，boundary : {r ∈ [0 , 1] | r ∈ Q}

d. interior : {(x, y) ∈ R^2 | 0 < x < 1}

　closure : {(x, y) ∈ R^2 | 0 ≤ x ≤ 1}

　boundary : {(x, y) ∈ R^2 | x = 0 or 1}

e. interior : φ，closure : {x ∈ R^n | ||x|| = 1}，boundary : {x ∈ R^n | ||x|| = 1}

3.

(a) no accumulation points

(b) {x | x ∈ R^2}

(c) {(x , 0) ∈ R^2 | x ∈ [0 , infinity)∩Q}∪{(0 , 1/n) ∈ R^2 | n is an integer}

(d) {(x , 0) ∈ R^2 | x = 0 or x = 1/n , n is an integer}

2011-01-18 22:35:31 補充：

sorry,

3(c) 應該是

{(x , 1/n) ∈ R^2 | x ∈ [0 , infinity)∩Q , n is an integer}∪{(0 , 0)}

Answer 2

1 (e)

即使 A 是 closed 也未必成立

例如單點

2011-01-18 13:47:19 補充：

3(c) 也有一點問題 ^_^

2011-01-18 23:50:21 補充：

x ∈ R 即可吧!!

例如

(1/1,1/1), (14/10,1/10), (141/100,1/100), (1414/1000,1/1000),

(14142/10000,1/10000), (141421/100000, 1/100000) ....

accumulation point 是 (sqrt(2), 0)

2011-01-19 06:47:42 補充：

3(c) {(x,0)|x∈ R} 比較合理