# Definition:Maximal/Set

(Redirected from Definition:Maximal Set)

## Definition

Let $S$ be a set.

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

Let $\mathcal T \subseteq \powerset S$ be a subset of $\powerset S$.

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

Then $T \in \mathcal T$ is a maximal set of $\mathcal T$ if and only if $T$ is a maximal element of $\struct {\mathcal T, \subseteq}$.

That is:

$\forall X \in \mathcal T: T \subseteq X \implies T = X$