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 $0$.

The support set $[X]$ of any set $X$ is the image of the unique function into any singleton $X \to 1$, and is thus a subsingleton, and a singleton if $X$ is pointed. Note that this is different from the support of the unique function $X \to 1$, which is always the empty set $\emptyset$.

The support object of an object $A$ of a category is the image of its map to the terminal object. In the internal logic of a category, this corresponds to the propositional truncation.

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:

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

Related concepts

• compact support

• compactly supported cohomology

• split support

References

• Wikipedia, Support (mathematics)