  We present statements equivalent to some fragments of the principle of
non-deterministic inductive definitions (NID) by van den Berg (2013), working
in a weak subsystem of constructive set theory CZF. We show that several
statements in constructive topology which were initially proved using NID are
equivalent to the elementary and finitary NIDs. We also show that the finitary
NID is equivalent to its binary fragment and that the elementary NID is
equivalent to a variant of NID based on the notion of biclosed subset. Our
result suggests that proving these statements in constructive topology requires
genuine extensions of CZF with the elementary or finitary NID.
