If three closed sets cover a triangle and respectively contain its three opposite sides, then their triple intersection is nonempty.
Normalized distances to three closed sets give continuous barycentric coordinates; a covering forces at least one coordinate to vanish, placing the image on the simplex boundary.
A closed-cover intersection statement for a simplex is equivalent to the nonexistence of a boundary-valued map that preserves every face.

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  1. Knaster-Kuratowski-Mazurkiewicz lemma
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