In a recent paper, Ziga Virk defined a type of continuous map which preserves extension properties. We generalize this notion and call such maps extensional maps. In this paper we will establish many of the basic properties of extensional maps. We will then show that extensional maps are preserved by the limit of an inverse system. Finally, there is a generalization of inverse systems called approximate inverse systems, due to Marde si c and Rubin. We will prove several new results concerning these approximate systems, and then show that extensional maps are preserved by the limit of an approximate system as well.