# Definition:Minimally Closed Class

## Definition

Let $A$ be a class.

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

### Definition 1

$A$ 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 no proper subclass of $A$ containing $b$ is closed under $g$. |

### Definition 2

$A$ 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

- 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.