On disjoint dynamical properties and Lipschitz-free spaces

The paper’s Theorem 2.7 proves the lifting of disjoint A-transitivity from an invariant linearly dense subspace Z⊂X to the whole space X but assumes A is a filter and 0∈Z, using a construction that requires finite intersections in A . The candidate solution gives a simpler argument: intersect the given open sets with Z, apply the hypothesis on Z to obtain a disjoint return set R∈A, and note R⊂d-NT(V0,V), so by upward-heredity (the only property needed from a Furstenberg family) we get d-NT(V0,V)∈A. This works without needing the filter property or 0∈Z, and aligns with the definitions of Furstenberg families and disjoint A-transitivity in the paper . Thus both are correct; the model’s proof is strictly sharper in assumptions.