\begin{definition}[Definition:Closed under Mapping/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 $\phi$''' {{iff}}:
:$\forall \family {s_i}_{i \mathop \in I} \in S^I \cap \Dom \phi: \map \phi {\family {s_i}_{i \mathop \in I} } \in S$
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 $\phi$''', {{iff}}:
:Whenever $\phi$ is defined for an $I$-indexed family from $S$, it maps that indexed family into $S$ again.
\end{definition}