CONTINUITY ON THE REAL LINE AND IN FORMAL SPACES
Erik Palmgren · 2005
Abstract Brouwer introduced his axioms for intuitionism to regain central results on continuity. Special axioms were avoided instead in Bishop's development of constructive analysis. Bishop in fact modified the definition of continuous function on the real numbers to mean uniformly continuous on each finite and closed interval. Though a successful move in the context of metric spaces, this seems to lead to difficulties when considering general spaces, in particular as the composition of two continuous functions needs not to be continuous. Though little emphasized, the continuous functions of the category of locales or formal spaces agree with Bishop's definition of continuous function on real numbers. Proving this within the framework of (Bishop) constructive mathematics is the purpose of the present chapter. The upshot is that for formal spaces, it is not necessary to adopt special axioms to obtain a good category.