\begin{definition}[Definition:Maximal/Set]
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 a '''maximal set''' of $\TT$ {{iff}} $T$ is a maximal element of $\struct {\TT, \subseteq}$.
That is:
:$\forall X \in \TT: T \subseteq X \implies T = X$
\end{definition}