I would like to share something I noticed on the definition of Maximal Positively Invariant Sets.
Definition 1. For a discrete-time system of the form $x_{k+1}=f(x_{k})$ (and $x_{k}\in \mathbb{X}\subseteq \mathbb{R}^n$ and $\mathbb{X}$ is a closed set), a set $\mathcal{O}\subseteq\mathbb{X}$ is called a Positively Invariant set if $x_{k+1}\in\mathcal{O}$ whenever $x_{k}\in\mathcal{O}$.
Definition 2a. The set $\mathcal{O}_\infty$ is called a Maximal Positively Invariant set is it is a Positively Invariant set and it contains all other positively invariant sets.
Definition 2b. On the other hand, the term maximal appears in partially ordered spaces like $(\mathbb{X},\subseteq)$ in a slightly different context. From that point of view, $\mathcal{O}_\infty$ is a Maximal Positively Invariant set if it is positively invariant and for every positively invariant set $\mathcal{O}_\infty'$ with $\mathcal{O}_\infty'\supseteq \mathcal{O}_\infty$, it is $\mathcal{O}_\infty'=\mathcal{O}_\infty$.
Are the two definitions equivalent?