  Exercise 00.0  Prove Fact 00.0.0: If   is a point in  -dimensional Euclidean space, and   is a finite set of hyperplanes, each of dimension less than  , then there is an   such that for every  ,   contains a point not on any hyperplane in  .  Exercise 00.0  Prove Fact 00.0.0: If   is a point in  -dimensional Euclidean space, and   is a finite set of hyperplanes, each of dimension less than  , none of which contains  , then there is an   such that   does not intersect any hyperplane in  .  Exercise 00.0  Prove Fact 00.0.0: A set of vertices   of a complex   forms a simplex if and only if the intersection of their open stars is non-empty