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.
Comments
A simplex is also a constituent of a simplicial complex, and a simplicial complex such that all subsets of its underlying subset are simplices is also called a simplex.
Simplex (abstract). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simplex_(abstract)&oldid=48708