3 CLOSED SETS, CLOSURE AND INTERIOR OF SETS
3.8 Theorem: Let (X, T) be a space and A ⊆ X, then
(i) X int(A) = X A
(ii) X A = int(X A).
Proof: This is clear from the definitions.
(i) x ∈ int(A) ⟺ ∃ U ∈ T such that x ∈ U ⊆ A. Thus x ∈ int(A) ⟺ ∀ U ∈ T, x ∈ U ⟹ U ∩ (X A) ≠ ∅.
(ii) We have x ∈ X A ⟺ ∃ U ∈ T with x ∈ U we must have U ∩ A = ∅. Thus, x ∈ X A ⟺ ∀ U ∈ T with x ∈ U such that U ∩ A = ∅ ⟺ U ⊆ X A.
Exercise: Write a complete proof.
