Final answer:
The question asks to prove that if the frontiers of sets A and B are disjoint in a topological space, then the interior of their union is equal to the union of their interiors. This involves showing that both Int(A) ∪ Int(B) ⊆ Int(A ∪ B) and Int(A ∪ B) ⊆ Int(A) ∪ Int(B), from which the equality follows.
Step-by-step explanation:
The student is asking to prove that if subsets A and B of a topological space X have disjoint frontiers (Fr A ∩ Fr B = ∅), then the interior of the union of these subsets is equal to the union of their interiors, i.e., Int(A ∪ B) = Int(A) ∪ Int(B).
We begin by noting that the interior of a set is the largest open set contained within it. Since open sets are closed under union, Int(A) ∪ Int(B) is an open set contained within A ∪ B. Hence, Int(A) ∪ Int(B) ⊆ Int(A ∪ B) directly from the definition of interior.
Conversely, consider any point x in Int(A ∪ B). By definition, there exists an open neighborhood around x entirely contained within A ∪ B. Without loss of generality, if x ∈ A, then x is also in Int(A) because x cannot be a frontier point of A due to the disjoint frontier condition. A similar argument holds if x ∈ B. Therefore, Int(A ∪ B) ⊆ Int(A) ∪ Int(B).
Combining both inclusions, we conclude that Int(A ∪ B) = Int(A) ∪ Int(B).