Knaster-Kuratowski-Mazurkiewicz lemma

For a simplex with vertices indexed by $0,\dots ,n$, let closed sets $F_i$ have the property that every face spanned by an index set $I$ lies in ${\bigcup }_{i\in I}{F}_i$. Then ${\bigcap }_{i=0}^n{F}_i$ is nonempty. The triangle KKM lemma is a two-dimensional covering formulation of this simplex-intersection principle.

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.