  The paper is a contribution to intuitionistic reverse mathematics. We work in
a weak formal system for intuitionistic analysis. The Principle of Open
Induction on Cantor space is the statement that every open subset of Cantor
space that is progressive with respect to the lexicographical ordering of
Cantor space coincides with Cantor space. The Approximate-Fan Theorem is an
extension of the Fan Theorem that follows from Brouwer's principle of induction
on bars in Baire space and implies the Principle of Open Induction on Cantor
space. The Principle of Open Induction in Cantor space implies the Fan Theorem,
but, conversely the Fan Theorem does not prove the Principle of Open Induction
on Cantor space. We list a number of equivalents of the Principle of Open
Induction on Cantor space and also a number of equivalents of the
Approximate-Fan Theorem.
