Definition:Greatest Set by Set Inclusion

Definition
Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$.

Let $\struct {\TT, \subseteq}$ be the ordered set formed on $\TT$ by $\subseteq$ considered as an ordering.

Then $T \in \TT$ is the greatest set of $\TT$ $T$ is the greatest element of $\struct {\TT, \subseteq}$.

That is:
 * $\forall X \in \TT: X \subseteq T$

Also see

 * Greatest Set is Unique
 * Greatest Set may not Exist


 * Definition:Smallest Set by Set Inclusion