0Simplex

0-simplex, in the context of simplicial geometry, is the simplest kind of simplex: a single point. It is a 0-dimensional object and can be described as the convex hull of one vertex. In a simplicial complex, a 0-simplex corresponds to a vertex, and it has no edges or higher-dimensional faces; its only face is the empty set, sometimes referred to as the (-1)-simplex. The geometric realization of a 0-simplex is just a single point.

More generally, the standard n-simplex Δ^n is the convex hull in R^{n+1} of the standard basis vectors e_0,...,e_n, equivalently the set of points (t0,...,tn) with ti ≥ 0 and sum ti = 1. When n=0, Δ^0 is a point; for n≥1, Δ^n has faces that are the Δ^i for i=0,...,n-1. A 0-simplex is thus the building block of all higher simplices.

In algebraic topology, 0-simplices form the basis of the chain groups C_0; oriented 1-simplices have boundaries equal to the difference of their endpoints, mapping to 0-simplices. The space consisting of a single 0-simplex is contractible and has trivial higher homology; its unreduced H_0 is Z, reflecting its single connected component, while the reduced tilde H_0 is zero.

Notation and usage: 0-simplex is sometimes denoted Δ^0. It is synonymous with a single vertex or, in simple terms, a point. Understanding 0-simplices is essential for building and analyzing simplicial complexes, meshes, and various constructions in topology and computational geometry.