Definition:Special Set

Definition
Let $g$ be a progressing mapping.

Let $S$ and $x$ be sets.

We say that:
 * $S$ is special for $x$


 * $(1): \O \in S$
 * $(2): S$ is closed under $g$ relative to $x$
 * $(3): S$ is closed under chain unions
 * $(3): S$ is closed under chain unions

Also known as
Instead of $S$ is special for $x$, we can say $S$ is $x$-special.