Definition:Closed under Mapping

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Definition

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

Let $S' \subseteq S$.


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

$f \left[{S'}\right] \subseteq S'$

where $f \left[{S'}\right]$ is the image of $S'$ under $f$.


Arbitrary Product

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.


Also see