The extensions of internal function approximation theorem and overflow theorem and theirs applicitaons

In the κ-saturated nonstandard model,the internal function approximation theorem was extended at first.By this theorem,the famous Ascoli theorem was proved.Then the overflow and underflow theorems were extended to a directed set.As their applications,some properties of monads on topological space were obtained and a proof of the infinitesimal prolongation theorem on nets was presented.