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 ? is a set of finite sets of vertices, known as faces , such that
::Given a simplicial complex ? as above, we define a (point set) topology on ? by declaring a subset be closed if and only if ? is a simplicial complex, i.e.
::This is the Alexandrov topology on the poset of faces of ?.

The order complex associated to a poset (S, <=) has the set S as vertices, and the finite chains of (S, <=) as faces. The poset topology associated to a poset (S, <=) is then the Alexandrov topology on the order complex associated to (S, <=).