# Definition:Closed under Mapping

## Definition

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

Let $S' \subseteq S$.

Then $S'$ is **closed under $f$** if and only if:

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

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

That is:

- $x \in S' \implies \map f x \in S'$

### Arbitrary Product

Let $\phi: X^I \to T$ be a mapping or a partial mapping, taking $I$-indexed families as arguments.

Denote with $\Dom \phi$ the domain of $\phi$ (if $\phi$ is a mapping, this is simply $X^I$).

A set $S$ is **closed under $\phi$** if and only if:

- $\forall \family {s_i}_{i \mathop \in I} \in S^I \cap \Dom \phi: \map \phi {\family {s_i}_{i \mathop \in I} } \in S$

Phrased in terms of image of a mapping, this translates to:

- $\map \phi {S^I \cap \Dom \phi} \subseteq S$

Thus, in words, $S$ is **closed under $\phi$**, if and only if:

- Whenever $\phi$ is defined for an $I$-indexed family from $S$, it maps that indexed family into $S$ again.

## 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

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications: Definition $4.1$