Order complex

Summary

The order complex of a partially ordered set $P$ is a simplicial complex with the elements of $P$ as vertices. Its faces are the finite chains of $P$. It turns a poset into a topological space, its geometric realization. So one can speak of the homotopy type and the homology of a poset. This is the basis of poset topology. For example, take an open interval of a finite poset. By a theorem of Philip Hall, the reduced Euler characteristic of its order complex equals the value of the Möbius function on that interval. The order complex of the face poset of a simplicial complex is its barycentric subdivision.