In this article, we study space-like and time-like surfaces in a Robertson-Walker space-time, denoted by \(L^4_1(f,c)\) , having positive relative nullity. First, we give the necessary and sufficient conditions for such space-like and time-like surfaces in \(L^4_1(f,c)\) . Then, we obtain the local classification theorems for space-like and time-like surfaces in \(L^4_1(f,0)\) with positive relative nullity, where the second factor is 3-dimensional Euclidean space. Finally, we consider the space-like and time-like surfaces in \(\mathbb {E}^1_1\times \mathbb {S}^3\) and \(\mathbb {E}^1_1\times \mathbb {H}^3\) with positive relative nullity. These are the special spaces of \(L^4_1(f,c)\) when the warping function f is a constant function, with \(c=1\) for \(\mathbb {E}^1_1\times \mathbb {S}^3\) and \(c=-1\) for \(\mathbb {E}^1_1\times \mathbb {H}^3\) .
