The continuity in the inductive sets: basic concepts and auxiliary results (Q2703275)

The partially ordered set \(\langle D,\preceq\rangle\) is called an inductive set if there is a minimal element in \(\langle D,\preceq\rangle\) and any non-empty chain has a supremum. The function \(f:D\to D'\) (where \(D, D'\) are inductive sets) is called continuous if for any non-empty chain \(X\) the following equality is fulfilled: \(f(\bigsqcup_{D}X)=\bigsqcup_{D'}f[X]\), where \(f[X]\) is a total image of \(X\). Scott's topology is introduced on the inductive sets and the following result is proved. The function \(f\) is continuous under the given definition if and only if \(f\) is continuous in Scott's topology. Cartesian products and extended Cartesian products of inductive sets are considered. The author proves that the Cartesian product of inductive sets is an inductive set.