In many an example of a function space whose topology is the topology of almost uniform convergence it is observed that the same topology is obtained in a natural way by considering pointwise convergence of extensions of the functions on a larger domain [1; 2]. This paper displays necessary conditions and sufficient conditions for the above situation to occur. Consider a linear space G(S, F) of functions with domain S and range in a real or complex locally convex linear topological space F. Assume that there are sufficient functions in G(S, F) to distinguish between points of S. Let S, denote the closure of the image of S in the cartesian product space X { g(S): gCG(S, F) }. Theorems 4.1 and 4.2 of reference [2] give the following theorem.