Math 2111 Advanced Calculus (I)
Homework 10-1 Hand in Problems:
2, 3, 4, 5 Lecture Note: 1, 2, 4
1. LetA be a subset in a metric space (M, d) andB ⊆M is open relative toA. Prove that A\B is closed relative to A.
2. Let A1 = (0,1) and A2 = (1,2]. Suppose that A=A1∪A2.
(a) Prove thatA1 and A2 are both closed and open relative to A.
(b) If A2 is replaced by [1,2], determine whether A1 and A2 are still closed and open relative to A.
3. Let A be a connected subset in a metric space. If K ⊆A is a nonempty compact subset which is open relative to A, prove that A is compact.
4. Prove that the following maps are continuous.
(a) A norm k · kdefined on a vector space V is continuous on V.
(b) Let (M, d) be a metric space and A ⊆ M be a nonempty set. Then the distance function f(x) :=d(x, A) is continuous on M.
5. (a) Use Problem 4(a) to prove that, forr >0, the set n x∈V
kxk> ro
is open inV. (b) Let (M, d) be a metric space andA⊆M. Forr >0, defineAr:=
n x∈M
d(x, A)≤r o
. Use Problem 4(b) to prove that Ar is closed in M.
Lecture Note:
• (Page 102)
1. Problem 2.36
• (Page 122) 2. Problem 3.2 3. Problem 3.3 4. Problem 3.4
