 There is a close connection between the bar principle and the neighborhood functions mentioned in the section on continuity axioms. Let \(\mathcal{IK}\) be the inductively defined class of neighborhood functions, consisting of all constant non-zero sequences \(\lambda m.n+1\), and such that if \(f(0)=0\) and \(\lambda m.f(x\cdot m)\in \mathcal{IK}\) for all \(x\), then \(f \in \mathcal{IK}\). The statement \(\mathcal{K}=\mathcal{IK}\), that is, the statement that the neighborhood functions can be generated inductively, is equivalent to BID. 