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A [[Topological space|topological space]]  $  | A | $
whose points are non-negative functions  $  \phi :  A \rightarrow \mathbf R $
on a finite set  $  A $
satisfying  $  \sum _ {a \in A }  \phi ( a) = 1 $. 
The topology on  $  | A | $
is induced from  $  \mathbf R  ^ {A} $, 
the space of all functions from  $  A $
into  $  \mathbf R $. 
The real numbers  $  \phi ( a) $
are called the barycentric coordinates of the point  $  \phi $, 
and the dimension of  $  | A | $
is defined as  $   \mathop{\rm card} ( A) - 1 $. 
In case  $  A $
is a linearly independent subset of a Euclidean space,  $  | A | $
is homeomorphic to the convex hull of the set  $  A $(
the homeomorphism being given by the correspondence  $  \phi \mapsto \sum _ {a \in A }  \phi ( a) \cdot a $). 
The convex hull of a linearly independent subset of a Euclidean space is called a Euclidean simplex.

For any mapping  $  f:  A \rightarrow B $
of finite sets, the formula  $  (| f | \phi ) ( b) = \sum _ {f ( a) = b }  \phi ( a) $, 
$  b \in B $, 
defines a continuous mapping  $  | f |:  | A | \rightarrow | B | $, 
which, for Euclidean simplices, is an affine (non-homogeneous linear) mapping extending  $  f $. 
This defines a functor from the category of finite sets into the category of topological spaces. If  $  B \subset  A $
and  $  i:  B \rightarrow A $
is the corresponding inclusion mapping, then  $  | i | $
is a homeomorphism onto a closed subset of  $  | A | $, 
called a face, which is usually identified with  $  | B | $. 
Zero-dimensional faces are called vertices (as a rule, they are identified with the elements of  $  A $).

A topological ordered simplex is a topological space  $  X $
together with a given homeomorphism  $  h:  \Delta  ^ {n} \rightarrow X $, 
where  $  \Delta  ^ {n} $
is a [[Standard simplex|standard simplex]]. The images of the faces of  $  \Delta  ^ {n} $
under  $  h $
are called the faces of the topological ordered simplex  $  X $. 
A mapping  $  X \rightarrow Y $
of two topological ordered simplices  $  X $
and  $  Y $
is said to be linear if it has the form  $  k \circ F \circ h  ^ {-} 1 $, 
where  $  k $
and  $  h $
are the given homeomorphisms and  $  F $
is a mapping  $  \Delta  ^ {n} \rightarrow \Delta  ^ {n} $
of the form  $  | f | $.

A topological simplex (of dimension  $  n $) 
is a topological space  $  X $
equipped with  $  ( n + 1)! $
homeomorphisms  $  \Delta  ^ {n} \rightarrow X $(
that is, with  $  ( n + 1)! $
structures of a topological ordered simplex) that differ by homeomorphisms  $  \Delta  ^ {n} \rightarrow \Delta  ^ {n} $
of the form  $  | f | $, 
where  $  f $
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|Simplicial set]]) and distinguished subsets of simplicial schemes (cf. [[Simplicial scheme|Simplicial scheme]]) are also referred to as simplices.

====Comments====
A simplex is also a constituent of a [[Simplicial complex|simplicial complex]], and a simplicial complex such that all subsets of its underlying subset are simplices is also called a simplex.