Definition of globally uniformly asymptotically stable

Question

In the book "Nonlinear systems" (Khalil, 3rd edition, page 150), an equilibrium is said to be globally uniformly asymptotically stable if

1. it is uniformly stable,
2. $\delta(\epsilon)$ can be chosen to satisfy $\lim_{\epsilon\to\infty}\delta(\epsilon)=\infty$,
3. for each pair of positive numbers $\eta$ and c, there is $T(\eta,c)>0$ such that $$||x||<\eta, \forall t\geq t_0+T(\eta,c), \forall ||x(t_0)||<c.$$

I have a question about the second condition. Why shall we need it? Is it necessary?