In this note our principal interest is in using nets to give spaces of non-absolutely convergent integrals as extensions of the spaces of absolutely convergent Riemann and Lebesgue integrals. For this purpose we develop a general theory of extensions, by nets, of functions defined on the open intervals with closures in the complement of a fixed closed set, the nets being directed by inclusion for finite disjoint collections of such intervals. Two cases are considered leading to open extension (OE-) and conditional open extension (COE-) nets, the latter being subnets of the former. Necessary and sufficient conditions for the convergence of the OE- and COE-nets are given, those for the COE-nets being similar to conditions that arise in the definition of the restricted Denjoy integral. Properties of inner continuity, weak additivity and the existence of a continuous integral are defined and studied. These relate to the more specialized nets that are suitable for the extension of integrals.