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.