Definition 40.1.2 (Invariant).label Let $G \curvearrowright X$ be a $G$-flow and $A \subset X$, then the following are equivalent:
- (1)
$GA \subset A$.
- (2)
$gA = A$ for all $g \in G$.
If the above holds, then $A$ is an invariant subset of $G$.
Proof. (1) $\Rightarrow$ (2): Let $g \in G$, then $gA \subset A$ and $g^{-1}A \subset A$ by (1). In which case, $A = gg^{-1}A \subset gA$, so $gA = A$.$\square$
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