We will discuss the usefullness of considering the weak topology in the dual of a Banach space, in particular for a space of the form $C(K_A)$, where $K_A$ is the Stone space of a Boolean algebra $A$ and of comparing it with the weak* topology. This naturally leads to considering the weak compactness and to introducing the Grothendieck property. We will also discuss some aspects of the interplay between compact spaces (or Boolean algebras) and the Banach spaces of continuous function defined on them. The talk should motivate some notions which will appear during the following talk by Zdenĕk Silber on some dichotomy under Martin's axiom.