# Definition:Closed under Mapping/Arbitrary Product

## Definition

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

Denote with $\operatorname{dom} \phi$ 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:

$\forall \left\langle{s_i}\right\rangle_{i \in I} \in S^I \cap \operatorname{dom} \phi: \phi \left({\left\langle{s_i}\right\rangle_{i \in I}}\right) \in S$

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

$\phi \left({S^I \cap \operatorname{dom} \phi}\right) \subseteq S$

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.