nat 發問時間: 科學數學 · 1 0 年前

[數學]高微的問題 急

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}

2 個解答

評分
  • 1 0 年前
    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)}

  • linch
    Lv 7
    1 0 年前

    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

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