A simplex is the Convex hull of a set of points which are not coplanar. If the points are $v_0,..,v_n$, the simplex is $n$-dimensional and is denoted $[v_0,..,v_n]$.

The standard $n$-dimensional simplex $\Delta^n$ is the Convex hull of the set of points corresponding to basis vectors ($(0,...,1,...,0)$) in $\mathbb{R}^{n+1}$.

If we delete one of the n + 1 vertices of an n simplex [v 0 , ··· , v n ] , then the remaining n vertices span an (n − 1) simplex, called a face of [v 0 , ··· , v n ]

The union of all the faces of ∆ n is the boundary of ∆ n , written ∂∆ n . The open simplex $\Delta^{\circ n}$ (actually $\circ$ should be on top..) is $\Delta^n - \partial \Delta^n$, the interior of $\Delta^n$.

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In Delta-complexes, we consider simplices with a given ordering of vertices.