BROUWER’S FAN THEOREM AND CONVEXITY

In the framework of Bishop’s constructive mathematics we introduce co-convexity as a property of subsets B of {0, 1}∗, the set of finite binary sequences, and prove that co-convex bars are uniform. Moreover, we establish a canonical correspondence between detachable subsets B of {0, 1}∗ and uniformly continuous functions f defined on the unit interval such that B is a bar if and only if the corresponding function f is positive-valued, B is a uniform bar if and only if f has positive infimum, and B is co-convex if and only if f satisfies a weak convexity condition.