Math 205a Fall Term 2002

Assignment 4

1. Let A = (a_{jk})_{j,k∈ℕ} be a double array of real numbers and let

d = (d_{1}, d_{2}, d_{3}, ⋯) = (a_{11}, a_{21}, a_{12}, a_{31}, a_{22}, a_{13}, ⋯)

be the sequence obtained by concatenating the finite diagonals

d_{m} = (a_{m1}, a_{m−1 2}, ⋯, a_{1m}) , m ∈ ℕ .

Show that any limit point of any row A_{j•} = (a_{jk})_{k∈ℕ} or column A_{•k} = (a_{jk})_{j∈ℕ} of A is a also a limit point of the sequence d. Do we obtain all limit points of d this way?

2. Let A ⊂ ℝ. We say that B ⊂ A is open in A, or, concisely B ⊂^{o} A, iff there is an open set B̃ ⊂ ℝ with

B = A ∩ B̃ .

Show that

(i) ∅, A ⊂^{o} A.

(ii) If B_{j} ⊂^{o} A for j ∈ ℕ, then ⋃_{j∈ℕ}B_{j} ⊂^{o} A.

(iii) If B_{j} ⊂^{o} A for j = 1, …, m (m ∈ ℕ), then ⋂_{1≤j≤m}B_{j} ⊂^{o} A.

Is [0, 1/2) open in [0, 1]? What about (0, 1/2]? Is {0} open in ℕ? Is it open in ℚ?

3. Let x ∈ ℝ^{ℕ} and let

X = {y ∈ ℝ | y = x_{j} for some j ∈ ℕ} .

What is the relation between the limit points of the sequence x and those of the set X?

4. Let A ⊂ ℝ. The sets \overline{A}, A^{∘} and LP (A) were defined in class. Let, in

addition, ∂A = \overline{A} \ A^{∘}. Prove or disprove the following:

A^{∘} ⊂ \overline{A} , \overline{A} = LP (A) ,

LP (A) ⊂ A , LP (LP (A)) ⊂ LP (A) ,

LP (A) ⊂ LP (LP (A)) , \overline{∂A} = ∂A ,
\overline{A} = LP (A) ∪ A , ∂(∂A) = ∂A .

The Homework is due Friday, November 1