Definition:Minimally Closed Class/Definition 2

Definition

Let $A$ be a class.

Let $g: A \to A$ be a mapping on $A$.

$A$ is minimally closed under $g$ with respect to $b$ if and only if:

\((1)\)	$:$							$A$ is closed under $g$
\((2)\)	$:$							There exists $b \in A$ such that every subclass of $A$ containing $b$ which is closed under $g$ contains all the elements of $A$.

Also see

• Equivalence of Definitions of Minimally Closed Class

• Results about minimally closed classes can be found here.

Linguistic Note

The term minimally closed class was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ in accordance with the concept of a minimally inductive class.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.

Both minimally closed class and minimally inductive class are discussed in Set Theory and the Continuum Problem, revised ed. by Raymond M. Smullyan and Melvin Fitting of $2010$.

While the minimally closed class is suggested as a generalisation of the minimally inductive class in an optional section backed up with an exercise, it is not actually named there as such.

The name was coined in order to provide a convenient tag, as a description is unwieldy.