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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?)

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    $\begingroup$ Cover a circle with three arcs … $\endgroup$ – Daniel Fischer Oct 19 '16 at 21:14
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    $\begingroup$ Let $A = [2,5]$, $B = [4,7]$ and $C = [0,3]\cup[6,9]$. $\endgroup$ – Joey Zou Oct 19 '16 at 21:15
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    $\begingroup$ Use pairs. $ $ $ $ $\endgroup$ – Did Oct 26 '16 at 8:12
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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.

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