Carrier (mathematics)

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In the mathematics of the designated carrier (Engl. Support ) usually the closure of the non-zero set of a function or other objects.


Bearer of a function

The bearer of is usually referred to as or .

Be a topological space and a function. The carrier of then consists of the closed envelope of the non-zero set of , formally:

Carrier of a distribution

Be an open subset of and a distribution. One says that a point belongs to the bearer of , and writes if a function exists for every open environment of .

If there is a regular distribution with continuous f , then this definition is equivalent to the definition of the carrier of a function (the function f ).


Is with , then is , for the non-zero set of is whose closure is integer . The same is true for any polynomial function except the null function .

Is with , if , else , then is the amount .

Is the characteristic function of , if , and , if , then is the carrier , i.e. the closure of .

Be an open subset of the . The set of all continuous functions from to with compact support forms a vector space , which is denoted by.

The set of all smooth (infinitely often continuously differentiable) functions with compact support plays a major role in the theory of distributions as the set of “test functions” .

The delta distribution has the carrier , because the following applies: if it is over , then it is .

Sheaf theory

Let it be a sheaf of Abelian groups over a topological space .

Carrier of a cut

For an open subset and a section , the end of the set of those points for which the image of in the stalk is not equal to zero is called the carrier of , usually referred to as or .

In particular, the term carrier of a vector field defined on a manifold denotes the termination of the set of points in which the vector field is not zero.

The carrier of a cut is always completed by definition.

Carrier of a sheaf

The bearer of itself is the set of points for which the stalk is non-zero.

The carrier of a sheaf is not necessarily closed, but the carrier of a coherent module sheaf is .


  • Roger Godement: Théorie des faisceaux . Hermann, Paris 1958.

Individual evidence

  1. The spelling may be confused with the trace of a square matrix, which is called trace in English .