Point-Set Apartness [chapter]

Douglas S. Bridges, Luminiţa Simona Vîţă
2011 Apartness and Uniformity  
Synopsis We first introduce the notion of a (pre-)apartness between points and subsets in an abstract space X, and derive some elementary properties from our axioms. Each point-set pre-apartness gives rise to a topology-the apartness topology-on X, and to several constructively distinct continuity properties, which are explored in Section 2.3. Limits and the Hausdorff property are discussed in Section 2.4, and product pre-apartness spaces in Section 2.5. In the final section we discuss the role
more » ... we discuss the role of impredicativity in our theory. Pre-apartness Throughout this chapter our basic structure will be an inhabited set X equipped with an inequality relation =, which in this context can also be called a pointpoint apartness. A subset S of X has two natural complementary subsets: the logical complement ¬S ≡ {x ∈ X : ∀ y∈S ¬ (x = y)} , ,
doi:10.1007/978-3-642-22415-7_2 fatcat:2ypa5axwgrcb3j7slejdwodr7u