Lecture Note:

Math 2111 Advanced Calculus (I)

Homework 5-1 Hand in Problems:

2, 3(b), 4(a)(d)(e), 5, 6, Lecture Note: 2, 3, 4

1. Let d₁ and d₂ be equivalent metrics on M . Prove that if U is open in (M, d₁) then U is open in (M, d2).

2. Let (M, d) be a metric space. Suppose that A is open and B is closed.

(a) Prove that every r-ball B(x, r) is open.

(b) Prove that the set {y ∈ M | d(x, y) ≤ r} is closed.

(c) It is easy to see that A\B = A ∩ B^C is open and B\A = B ∩ A^C is closed. Use the definitions of open sets and closed sets to prove the same results.

3. Let (M, d) be a metric space.

(a) Prove that a set consisting of a single point is closed.

(b) Use the definition of closed sets to prove that a set consisting of finitely many points is closed.

4. For the given set A, find ˚A, A⁰ and ¯A. Use the definitions of interior point, accumulation point and limit point to check your answers.

(a) A = Q ∩ [0, 1].

(b) A = {1

n | n ∈ N}.

(c) A = [0, 1] × (0, 1).

(d) A = {(x, y) ∈ R² | 0 < x ≤ 1}.

(e) A = {(x, y) ∈ R² | x = y}.

(f) Let f (x) = sin 1

x
on (0, 1). A = Graph(f ) = {(x, y) ∈ R² | y = f (x), x ∈ (0, 1)}.

5. Let A₁, A₂, A₃, · · · be subsets of a metric space.

(a) If B_n=Sn

i=1A_i, prove that B_n=

n

[

i=1

A_i.

(b) If B =S∞

i=1A_i, prove that B ⊇

∞

[

i=1

A_i.

(c) Show, by an example, that the inclusion “⊇” in (b) can be %.

6. Let (M, d) be a metric space and A ⊆ M . (a) Prove that A⁰ are closed.

(b) Prove that A⁰ = (A)⁰.

(c) Determine whether A⁰ = (A⁰)⁰.

Lecture Note:

• (Page 54)

1. Proposition 2.9

• (Page 94) 2. Problem 2.1 3. Problem 2.3 4. Problem 2.4