Consider a set $\mathcal{C}$ and initial condition $x(0)=x_{0}$. $\mathcal{C}$ is forward invariant if $$x_{0}\in \mathcal{C} \implies x(t)\in \mathcal{C},\,\forall t\ge 0$$i.e. if we start in $\mathcal{C}$ we stay in $\mathcal{C}$.

The system $$\dot{x}=f_{\text{cl}(x)}:=f(x)+g(x)k(x)$$where $k(x)=u$ is the feedback controller, is safe with respect to the set $\mathcal{C}$ if the set $\mathcal{C}$ is .