Discussion
Let A and B be two non-empty disjoint subsets of \(\mathbf{R}\) and let \(d(A, B) = \inf\{|a - b| : a \in A, b \in B\}\). Then \(d(A, B) \gt 0\) if :
Answer: C
\(d(A, B) \gt 0\) is guaranteed when A is closed and B is compact.
The function \(b \mapsto d(b, A)\) is continuous, and it is positive at each \(b \in B\) because \(b \notin A\) and A is closed. A continuous positive function on a compact set attains a positive minimum, so \(d(A, B) \gt 0\).
The other conditions are not enough. Open sets: A = (0, 1), B = (1, 2) are disjoint with distance 0. Closed sets: A = \(\mathbf{N}\) and \(B = \{n + 1/n : n \ge 2\}\) are closed and disjoint but get arbitrarily close. A singleton: A = {0}, B = (0, 1) gives distance 0.
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