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.We work in a weak subsystem of CZF, called the elementary constructive set theory ECST [4], where none of the known fragments of the NID principle seems to be derivable.The language of ECST contains variables for sets and binary predicates = and ∈. The axioms and rules of ECST are the axioms and rules of intuitionistic predicate logic with equality, and the following set-theoretic axioms:Pairing: ∀a∀b∃y∀u (u ∈ y ↔ u = a ∨ u = b) .Union: ∀a∃y∀x (x ∈ y ↔ ∃u ∈ a (x ∈ u)) .
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