Face poset and order complex

For an abstract simplicial complex $K$, its face poset is $F(K)=K\setminus \{\mathrm{\varnothing }\}$ ordered by inclusion. For a poset $(P,\le )$, its order complex $\mathrm{\Delta }P$ has vertex set $P$ and faces all finite chains in $P$, including the empty chain. Here a chain means a subset in which every two elements are comparable.

Inclusion is reflexive, antisymmetric and transitive, as required by Partial order and partially ordered set. Every subset of a finite chain is a finite chain, and each singleton is a chain, so this satisfies An abstract simplicial complex. If $P=\mathrm{\varnothing }$, then $\mathrm{\Delta }P=\{\mathrm{\varnothing }\}$ has no vertices. If $P$ has a least element $p$, every face can be enlarged by $p$; this is a cone with a vertex, not the empty complex.

2.5.10, pp.51–52 (face-chain description); general-poset formulation is an explicit abstraction of the face-chain construction, not a quotation.