the topology of weak convergence

Weak convergence was defined as a notion of convergent sequences. But to use compactness, continuity of maps, and the full strength of point-set topology, you want it to be an honest topology, with open sets, not just a recipe for sequences. The topology of weak convergence is that topology on P(S): the coarsest topology making all the evaluation maps mu |-> integral f dmu continuous, for f ranging over C_b(S).

Concretely, a basic open neighbourhood of mu consists of all nu with |integral f_i dnu - integral f_i dmu| < epsilon for finitely many bounded continuous functions f_1, ..., f_k. This is precisely the weak-star topology that P(S) inherits as a subset of the dual of C_b(S). Its convergent sequences are exactly the weakly convergent ones, integral f dmu_n -> integral f dmu for all f in C_b(S), so for sequences it agrees with the definition you already know. When S is separable and metrizable this topology is itself metrizable, by the Levy-Prokhorov or bounded-Lipschitz metric, so sequences suffice and you need not fuss with nets.

Two cautions keep this honest. First, 'weak' here is the weak-star topology of functional analysis, not the weak topology of a Banach space; the abuse of language is universal in probability but worth knowing. Second, on a non-compact S the space P(S) is not compact, mass can escape to infinity, which is exactly why Prokhorov's theorem needs tightness to characterize relative compactness. When S is compact, P(S) is compact (by Banach-Alaoglu, since P(S) is a weak-star closed subset of the unit ball of the dual of C(S)), and life is much simpler.