A classical theorem states that any open set on the real line is a countable union of disjoint open invervals. Here the numerical content of this theorem is investigated with the methods of constructive topology.

Given a simplex S and a positive function δ on S, we show that there is a simplicial subdivision of S such that the diameter of each subdividing simplex is smaller that δ evaluated at some of its vertices.

In recent papers, Russell introduced the notions of functions of bounded kth variation (BVk functions) and the RSk integral. Das and Lahiri enriched Russell's works along with a convergence formula of RSk integrals depending on the convergence of integrands. In this paper a convergence theorem analogous to Arzela's dominated convergence theorem has been presented. An investigation to the convergence in kth variation has been made leading to some convergence theorems of RSk integrals depending on the convergence of integrators.

This paper introduces certain generalization of the notions of approximate limit, continuity and derivative and of absolute continuity, of real functions, leading to generalized integrals of Perron and Denjoy types comprising the AP-integral of Burkill (1931) and Sonouchi and Utagawa (1949) and the AD-integral of Kubota (1963), respectiely. The generalizations are all substantiated by appropriate examples.