Final answer:
A closed subset A and B of the real line R such that d(A, B) = 0 but A ∩ B is an empty set is given by the example: A = (-∞, 0) ∪ (1, ∞) and B = (0, 1).
Step-by-step explanation:
A closed subset A and B of the real line R such that d(A, B) = 0 but A ∩ B is an empty set is given by the example:
A = (-∞, 0) ∪ (1, ∞) and B = (0, 1).
To show that A and B are closed subsets, we can start by proving that their complements are open sets. The complement of A is (0, 1), which is an open set, and the complement of B is (-∞, 0) ∪ [1, ∞), which is also an open set. Therefore, A and B are closed subsets.
Next, we need to show that d(A, B) = 0. The distance between two sets A and B is defined as the infimum of the distances between any two points from A and B. In this case, the distance is 0 because for any positive number ε, we can always find points x in A and y in B such that the distance between x and y is less than ε. For example, we can choose x = 1 - ε/2 and y = 1 + ε/2.
Finally, we can see that A ∩ B is an empty set because there are no common points between A and B.