State the Definition of a Metric Space, Let (X, d) be a Metric Space: Mathematics Assignment, NTU, Singapore

Assignment Overview:

Question 1: (a) State the definition of a metric space.

(b) Let (X, d) be a metric space and f: A −→ X be an injective function from a set A into X. Define ˜d : A × A −→ [0,∞) by ˜d(x, y) = d(f(x), f(y)). Prove that ˜d is a metric on A.

(c) Give an example that illustrates an instance of (b).

Question 2: (a) Consider the following Condition (U) satisfied by a function d: X ×X −→ [0,∞).

∀x, y, z ∈ X. d(x, z) ≤ max{d(x, y), d(y, z)}.

Prove that Condition (U) implies (M4), the Triangle Inequality.

(b) Let X = 2ω be the set of all infinite binary sequences, i.e., sequences of zeros and ones. Define the function d : X × X −→ [0,∞) as follows:
d(x, y) = inf{2^(−n)| x =n y},
where x =n y means that the first n terms of x agree with those of y.
For example, let 0^ω:= 0000 · · · and 001^ω:= 00111 · · ·. Then 0^ω =2 001^ω but 0^ω 6~=3 001^ω.
Prove that (X, d) satisfies Condition (U) of (a), and hence show that (X, d) is a
metric space.

Question 3: (a) Let X1 be the set of all positive real numbers, and define for any x, y ∈ X1,
d1(x, y):= ln (y/x).
Prove that (X1, d1) is a metric space.

(b) Let X2 be the set of all m × n matrices over the field R of real numbers. Prove that (X2, d2) is a metric space, where for any A, B ∈ X2,
d2(A, B):= rank(A − B).
Here rank(M) denotes the rank of a matrix M ∈ X2.

Question 4: (a) Let (X, d) be a metric space, x0 ∈ X and r > 0.
Prove that the open ball B(x0; r) := {x ∈ X | d(x, x0) < r} is an open set.

(b) Is every open set of a metric space necessarily an open ball?
Justify your answer.

(c) Prove that a set C ⊆ X is closed if around each point not in C there exists a closed ball of non-zero radius which does not intersect C.

Question 5: Prove that the space C(R) of all continuous functions on R, equipped with the sup-metric: d(x, y):= sup t∈R |x(t) − y(t)| is not separable.
Hint: You may wish to consider the set
Λ := {f ∈ C(R) | f(n) = n or f(n) = 0 for all n ∈ N}.