Math 424A and 574A: HOMEWORK 3

Due on Wednesday, October 18, at the beginning of the class

2.15, 2.18, 2.19, 2.21, and the two problems below (turn in)

2.12, 2.13, 2.17, 2.20 (optional, do NOT turn in)


Problem 1 (turn in!) Decide whether the following sets in R1 are open, closed, or neither. Give a short justification.

(a) Z (the set of all integers)

(b) Q (the set of all rationals)

(c) {1/n: n &isin Z+}

(d) {1/n: n &isin Z+}&cup {0}

Problem 2 (turn in!) Decide whether the following sets in R2 are open, closed, or neither. Give a short justification. Note that (x,y) denotes a point in the plane (an ordered pair).

(a) {(x,y): x2+y2 >1}

(b) {(x,y): x2+y2 =1}

(c) {(x,2): 0 < x < 1}

(d) {(n,1/n): n &isin Z+}