Ideal Spaces and Maps: Ideal Aspects Over Real Topology

Abstract In this chapter ideal aspects, namely ideal spaces and ideal maps, are added to real ones, positive topologies. Ideal points gather all approximations of an unreal entity. Adding them to positive topologies gives ideal spaces. Examples include real numbers, Zariski topology, and Scott domains. Ideal spaces induce an ideal cover and positivity; their equality with real ones is called spatiality and reducibility, and represents significant principles. On trees, ideal points are paths, corresponding to Brouwer’s choice sequences, spatiality is bar induction, and reducibility is spread habitation. On rational numbers, ideal points of Dedekind–Joyal positive topology are Dedekind cuts; more effectively, real numbers are paths in a nested tree for signed digit representation. Ideal maps turn ideal spaces into a category ISpa. Its isomorphism with PTop provides a form of conservativity. Brouwer’s claim, that all mappings between real numbers are continuous, is proved assuming they are induced by a relation between observables.