nLab support

Idea

The term “support” means different things in different parts of mathematics:

In set theory

Given $f \colon X \to A$ a function to $A$ a pointed object with the point playing the role of a zero element, then the support of $f$ is the subobject of $X$ on which $f$ is non-zero.

In topology

In topology the support of a continuous function $f \colon X \to A$ as above is the topological closure of the set of points on which $f$ does not vanish:

$$Supp(f) = Cl(\{x \in X \vert f(x) \neq 0 \in A\}) \,.$$

If $Supp(f) \subset X$ is a compact subspace, then one says that $f$ has compact support.

In functional analysis

• support of a distribution

In measure theory

• Support of a measure
• Support of a valuation

In field theory

• spacetime support

In topos theory and type theory

support = (-1)-truncation

Related concepts

• compact support

• compactly supported cohomology

• split support

References

• Wikipedia, Support (mathematics)