attrativity, stability and global stability

Question

I am confusing in understanding the concept of attractivity and stability. What is the difference between attractivity and asymptotically stability? I am searching for an example that is attractive but it is not asymptotically stable. To infer globally uniformly asymptotically stable in non-autonomous is it necessary to check the following condition?: (i) for all $(\eta,c)>0$ there exist $T=T(\eta,c)>0$ such that $|x|<\eta$ for any $t\geq t_0+T$. My intuition is that when we have $\delta(\varepsilon)$ goes to infty when $\varepsilon$ goes to infty and there exists a radially unbounded function with negative definite derivative, condition (i) is satisfied. I mean in these conditions, there is no necessity to check condition (i), what's my mistake? There are many related questions but they do not answer my questions