Resolution of the uniform lower bound problem in constructive analysis

In a previous paper we constructed a full and faithful functor M from the category of locally compact metric spaces to the category of formal topologies (representations of locales). Here we show that for a real-valued continuous function f , M ( f ) factors through the localic positive reals if, and only if, f has a uniform positive lower bound on each ball in the locally compact space. We work within the framework of Bishop constructive mathematics, where the latter notion is strictly stronger than point-wise positivity. MSC: 03F55, 03F60, 06D22