nLab support of a function

This article is about support of a function. For other notions of support, see support.

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Definition

In set theory

Given a pointed set $A$ with specified element $0 \in A$, a set $X$, and a function $f \colon X \to A$, the support of $f$ is the subset of $X$ on which $f$ is not equal to $0$.

In constructive mathematics

In constructive mathematics, there are multiple notions of inequality, due to the failure of the double negation law. As a result, there are multiple notion of support of a function. Thus, we define the following:

Given a pointed set $A$ with specified element $0 \in A$, a set $X$, and a function $f \colon X \to A$, the support of $f$ is the subset of $X$ on which $f$ is not equal to $0$.

Given a pointed set $A$ with an tight apartness relation $\#$ and specified element $0 \in A$, a set $X$, and a function $f \colon X \to A$, the strong support of $f$ is the subset of $X$ on which $f$ is apart from $0$.

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.

See also

• compact support
• compactly supported cohomology
• support of a distribution

References

• Wikipedia, Support (mathematics)