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.
, 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.
The order complex associated to a poset (S, ≤) has the set S as vertices, and the finite chains of (S, ≤) as faces.