In mathematics, the poset topology associated to a poset (S, ≤) is the Alexandrov topology (open sets are upper sets) on the poset of finite chains of (S, ≤), ordered by inclusion. Let V be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices, known as faces , such that Given a simplicial complex Δ as above, we define a (point set) topology on Δ by declaring a subset 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.