Ideal-convergence classes

Let X be a non-empty set. We consider the class C consisting of triads (s,x,I), where s=(sd)d∈D is a net in X, x∈X, and I is an ideal of D. We shall find several properties of C such that there exists a topology τ for X satisfying the following equivalence: ((sd)d∈D,x,I)∈C, where I is a proper D-admissible ideal on D, if and only if (sd)d∈D I-converges to x relative to the topology τ.