Simplicial complex

Remarks

Example: on $V = \set{1, 2, 3}$, let $\Delta$ be the family of all subsets of $V$ except $V$ itself. A subset of a set with at most $2$ elements has at most $2$ elements, so clause 1 holds. Clause 2 holds, and $\Delta$ is the boundary of a triangle, of dimension $1$.

Non-example: on $V = \set{1, 2, 3}$, the family $\set{\emptyset, \set{1}, \set{2}, \set{3}, \set{1, 2}, \set{1, 2, 3}}$ fails clause 1, since $\set{1, 3} \subseteq \set{1, 2, 3}$ is missing.

A nonempty complex contains the empty face, of dimension $-1$. Some authors allow an infinite vertex set, keeping every face finite. A geometric realization places the vertices as affinely independent points in some $\R^N$ and replaces each face by the convex hull of its points.