DES - Definition of 'Closed Subset'

Closed Subset:

A subset $S\subseteq \mathcal{X}$ is closed if $q_{i,j}=0\forall i\in S,j\notin S$

Equivalently, there are no arcs starting from state in $S$ and ending in states not in $S$. Or, when the system enters the subset $S$, it remains in $S$ forever.

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~Example:

The subset $S=\{4,5,6\}$ is closed. The subset $S=\{1,2,3\}$ is not closed.

NOTE: Also the state set $\mathcal{X}=\{1,2,3,4,5,6\}$ is closed.