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.