Simplex (abstract)

A topological space whose points are non-negative functions on a finite set satisfying . The topology on is induced from , the space of all functions from into . The real numbers are called the barycentric coordinates of the point , and the dimension of is defined as . In case is a linearly independent subset of a Euclidean space, is homeomorphic to the convex hull of the set (the homeomorphism being given by the correspondence ). The convex hull of a linearly independent subset of a Euclidean space is called a Euclidean simplex.

For any mapping of finite sets, the formula , , defines a continuous mapping , which, for Euclidean simplices, is an affine (non-homogeneous linear) mapping extending . This defines a functor from the category of finite sets into the category of topological spaces. If and is the corresponding inclusion mapping, then is a homeomorphism onto a closed subset of , called a face, which is usually identified with . Zero-dimensional faces are called vertices (as a rule, they are identified with the elements of ).

A topological ordered simplex is a topological space together with a given homeomorphism , where is a standard simplex. The images of the faces of under are called the faces of the topological ordered simplex . A mapping of two topological ordered simplices and is said to be linear if it has the form , where and are the given homeomorphisms and is a mapping of the form .

A topological simplex (of dimension ) is a topological space equipped with homeomorphisms (that is, with structures of a topological ordered simplex) that differ by homeomorphisms of the form , where is an arbitrary permutation of the vertices. Similarly, a mapping of topological simplices is called linear if it is a linear mapping of the corresponding topological ordered simplices.

Elements of simplicial sets (cf. Simplicial set) and distinguished subsets of simplicial schemes (cf. Simplicial scheme) are also referred to as simplices.