# Definition:Transitive-Closed Class

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## Definition

Let $K$ be a class

Then $K$ is **transitive-closed** if and only if:

- every transitive subset of $K$ is an element of $K$.

## Also known as

Referred to in Raymond M. Smullyan and Melvin Fitting: *Set Theory and the Continuum Problem* (revised ed.) as a **$T$-closed class**.

## Also see

- Results about
**transitive-closed classes**can be found**here**.

## Linguistic Note

The term **Transitive-Closed Class** was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$.

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

The concept appears to have been coined by Raymond M. Smullyan and Melvin Fitting under the name **$T$-closed class** in their *Set Theory and the Continuum Problem, revised ed.* of $2010$ for the purpose of an exercise.

It is referred to on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a **transitive-closed class** in order to improve clarity and transparency.

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 2$ Ordinals and transitivity: Exercise $2.1$