1 To be Graded
Problem 1. Let (X, ρ) be a metric space, and f : [0,∞) → [0,∞) be an increasing concave
function such that f(r) = 0 if and only if r = 0. Prove that f ◦ρ is also a metric on X. Hint:
f being concave means that ∀p, q ∈ [0,∞) we have f(tp + (1 − t)q) ≥ tf(p) + (1 − t)f(q)
∀t ∈ [0, 1]. We can show here that f is subadditive in the following way: (i) take q = 0, we
can see that f(tp) ≥ tf(p); (ii) f(p) + f(q) = f(
p
p+q
(p + q)) + f(
q
p+q
(p + q)) ≥
p
p+q
f(p + q) +
q
p+q
f(p + q) = f(p + q).
Problem 2. Let X = R
2
. Define ρ1(x, y) ≡ |x1−y1|+|x2−y2|, ρ2(x, y) ≡
p
|x1 − y1|
2 + |x2 − y2|
2
and ρmax(x, y) ≡ max{|x1 − y1|, |x2 − y2|}. Prove that ρ1, ρ2 and ρmax are uniformly equivalent.
Problem 3. Let a, b ∈ R be two real numbers. Prove the following statements.
(i) The set X = (a, b), with the metric ρ(x, y) = |x − y|, is open;
(ii) The set X = [a, b], with the metric ρ(x, y) = |x − y|, is closed;
(iii) The set X = (a, b], with the metric ρ(x, y) = |x − y|, is neither open nor closed.
Problem 4. Let X be any non-empty set. We define ρ : X × X → [0,∞) as:
ρ(x, y) = ?
0, if x = y
1, if x 6= y
Then it can be shown that ρ is a metric on X. Therefore, (X, ρ) is a metric space. Such a
metric space is often called a discrete metric space. Let (X, ρ) be a discrete metric space.
Prove the following statements.
(i) An open ball in X is either a set with only one element (that is, a singleton) or all of X.
(ii) All subsets of X are both open and closed.
Problem 5. Let (X, ρ) be a metric space and S ⊆ X a subset. Denote by S the set of
points of closure of S. Prove that S is a closed set.
2 Reading Assignments
• Review Lecture Notes # 6 and # 7;
• Review Sections 2.2 and 3.1 of the textbook;
