# Maximal Positive Invariant Set — Some fine print

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?

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Is there some discrepancy between the two definitions after all? What if one may find two sets $\mathcal{O}_\infty'$ & $\mathcal{O}_\infty$ that are both maximal positively invariant (according to definition 2b this is feasible). Of course these two sets are neither a subset of the other one. It turns out that this is not the case and in fact this (pathological) pair of sets does not exist since unions of positively invariant sets are positively invariant. As a result, $\mathcal{O}_\infty\cup\mathcal{O}_\infty'$ would be a positively invariant set (hence, none of $\mathcal{O}_\infty$, $\mathcal{O}_\infty'$ is maximal).

Ergo: The two definitions are equivalent!

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