For any operator t on integer-valued functions, we say that #P is closed under t in context PF(#P) if, for every f in #P, t(f) belongs to PF^#P[1]. For several operators t, it is shown that the closure properties of #P under t in the above sense is closely related to the relationships between PF^#P[1] and higher classes such as PH^PP and PP^PP.