As for the linear case, compactness for the strong topology is very restrictive.  Since the beginning of the fixed point theory, weak-compactness offered an acceptable alternative in Banach spaces. But when we deal with metric spaces, this natural extension is no longer easy to implement. One has to go back to the linear case and investigate the weak-topology with a new eye.  In this talk, I will share some of the ideas of how to extend concepts of linear nature to nonlinear spaces, i.e., metric spaces.