if arbitrary intersection of compact sets is empty, then there exists at least two sets that are disjoint?

Question

if arbitrary intersection of compact set is empty, then there exists at least two sets that are disjoint?

Generally, I know the argument is false as nested intersection of open sets are empty, but there is not pair-wise disjoint. How about compact sets (closed and bounded in real line?)

Answer 1

As you already know from comments, in general the claim is false, but it holds for convex compact subsets of the real line as one-dimensional case of Helly’s theorem.