Definition:Minimal/Set

Definition
Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathcal T \subseteq \mathcal P \left({S}\right)$ be a subset of $\mathcal P \left({S}\right)$.

Let $\left({\mathcal T, \subseteq}\right)$ be the poset formed on $\mathcal T$ by $\subseteq$ considered as an ordering.

Then $T \in \mathcal T$ is the minimal set (or smallest set) of $\mathcal T$ iff $T$ is the minimal element of $\left({\mathcal T, \subseteq}\right)$.

That is:
 * $\forall X \in \mathcal T: X \subseteq T \implies X = T$