Definition:Smallest 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 smallest set of $\TT$ $T$ is the smallest element of $\struct {\TT, \subseteq}$.

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

Also see

 * Smallest Set is Unique
 * Smallest Set may not Exist


 * Definition:Greatest Set by Set Inclusion