MA2223: PROBLEM SHEET 2
1. Let (X, d) be a metric space and let A be a subset of X.
(a) Prove that the closure Ā is the intersection of all closed sets
which contain A.
(b) Using (a) show that the closure of A is closed.
2. Let (X, d) be a metric space and let a ∈ X. The closed ball with
centre a and radius r > 0 is defined as the set {x ∈ X : d(a, x) ≤ r}.
(a) Prove that this is a closed set.
(b) Prove that B(a, r) ⊆ {x ∈ X : d(a, x) ≤ r}.
(c) Prove that in Euclidean space, B(a, r) = {x ∈ X : d(a, x) ≤ r}.
(d) Give an example where B(a, r) 6= {x ∈ X : d(a, x) ≤ r}.
3. Let C[a, b] be the set of continuous real-valued functions on the
interval [a, b] together with the supremum metric
d(f, g) = sup |f (x) − g(x)|
x∈[a,b]
(a) Prove that if S is a subset of [a, b] then A = {f ∈ C[0, 1] :
f (x) = 0, ∀x ∈ S} is a closed set.
(b) Prove that if A is a subset of C[a, b] then {x ∈ [a, b] : f (x) =
0, ∀f ∈ A} is a closed set.
4. Let (X, d) be a metric space and let A and B be subsets of X.
Determine if the following identities are always true.
(a) (A ∪ B) = Ā ∪ B̄
(b) (A ∩ B) = Ā ∩ B̄
5. Let (X, d) be a metric space and let a ∈ X. Prove that the mapping
T : X → R, x → d(a, x) is continuous.
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