Resolution of the uniform lower bound problem in constructive analysis

Abstract

In a previous paper we constructed a full and faithful functor ℳ︁ 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, ℳ︁(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. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)