Definition:Closed under Mapping

Definition
Let $f: S \to T$ be a mapping.

Let $S' \subseteq S$.

Then $S'$ is closed under $f$ :


 * $f \sqbrk {S'} \subseteq S'$

where $f \sqbrk {S'}$ is the image of $S'$ under $f$.

That is:
 * $x \in S' \implies \map f x \in S'$

Also known as
A mapping $f$ such that $S'$ is closed under $f$ can itself be referred to as being closed in $S'$, but care needs to be taken to distinguish between this and the concept of a closed mapping in the context of topology.

Also see

 * Closed Algebraic Structure, an analogous concept in abstract algebra
 * Definition:Closed for Scalar Product