EXERCISE 2. 1. Let {(X

EXERCISE 2.

1. Let {(X_i, d_i) | i = 1,2, . . . , n} be a finite class of metric spaces. Show that each of the functions d and ˜d defined as follows is a metric on the product X₁×X₂× · · · ×X_n:

d(x,y) = max

1≤i≤nd_i(x_i, y_i) and ˜d(x,y) =

n

X

i=1

d_i(x_i, y_i)

where x = (x₁, x₂, . . . , x_n) and y = (y₁, y₂, . . . , y_n).

2. Let X be any nonempty set andd the discrete metric on X. Prove that every subset of X is open and every subset is closed.

3. Let d and d₁ be two metrics on R² defined by

d1(x,y) = max{|x1−y1|,|x2−y2|}

where x = (x1, x2) and y = (y1, y2), and d is the usual distance in R². Prove that a subset A ofR² is open in (R², d) if and only if it is open in (R², d1).

4. Prove that E is dense in X if and only if ¯E =X.

5. Prove that Q and R−Q are dense in R.

6. Let (X, d) be a metric space, A and B be subsets of X. Prove that (a) (IntA)∪(IntB)⊆Int(A∪B),

(b) (IntA)∩(IntB) = Int(A∩B)

where IntE denote the set of all interior points of a setE.

7. A point p is an element of ¯A if and only if for every positive real number r, N_r(p)∩A6=∅.

8. Prove thatX is connected if and only ifXhas exactly two subsets which are both open and closed inX.