A note on the non-forking-instances topology

The non-forking-instances topology (NFI topology) is a topology on the Stone space of a theory T that depends on a reduct (Formula presented.) of T. This topology has been used in [6] to describe the set of universal transducers for (Formula presented.) (invariants sets that translate forking-open sets in (Formula presented.) to forking-open sets in T). In this paper we show that in contrast to the stable case, the NFI topology need not be invariant over parameters in (Formula presented.) but a weak version of this holds for any simple T. We also note that for the lovely pair expansions, of theories with the weak non-finite cover property (wnfcp), the topology is invariant over (Formula presented.) in (Formula presented.).