Extended topology and systems

In this paper is indicated the possible utility of isotonic spaces as a background language for discussing systems. In isotonic spaces the basic duality between neighborhood and convergent first achieves a proper background permitting applications beyond the scope of topological spaces. A generalization of continuity of mappings based on ancestral relations is presented and this definition is applied to establish a necessary and sufficient condition that mappings preserve connectedness. Fortunately for systems theory, it is not necessary to have infinite sets or infinitary operators to apply definitions of neighborhood, convergents, continuity and connectedness.This work was supported in part by a grant from the National Science Foundation.