We study the boundary structure for w*-compact subsets of dual Banach spaces. To be more precise, for a Banach space X, 0 < ϵ < 1 and a subset T of the dual space X* such that ⋃ B(t,ϵ): t ∈ T contains a James boundary for $B_{X*}$ we study different kinds of conditions on T, besides T being countable, which ensure that $X*={\overline{spanT}}^{\left|\right|·\left|\right|}$. (SP) We analyze two different non-separable cases where the equality (SP) holds: (a) if $J:X\to 2^{B_{X*}}$ is the duality mapping and there exists a σ-fragmented map f: X → X* such that B(f(x),ϵ)...