  The $NFI$-topology, introduced in [S0], is a topology on the Stone space of a
theory $T$ that depends on a reduct $T^-$ of $T$. This topology has been used
in [S0] to describe the set of universal transducers for $(T,T^-)$ (invariants
sets that translates forking-open sets in $T^-$ 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 $T^-$ but a weak version of this holds
for any simple $T$. We also note that for the lovely pair expansions, of
theories with the \em wnfcp\em , the topology is invariant over $\emptyset$ in
$T^-$.
