# Definition:Closed under Mapping/Arbitrary Product

Jump to navigation
Jump to search

## Definition

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 see

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