 An abstract simplicial complex Δ is a set of finite sets of vertices, known as faces formula_1, such that
Given a simplicial complex Δ as above, we define a (point set) topology on Δ by declaring a subset formula_3 be closed if and only if Γ is a simplicial complex, i.e.This is the Alexandrov topology on the poset of faces of Δ.The order complex associated to a poset ("S", ≤) has the set "S" as vertices, and the finite chains of ("S", ≤) as faces.