Generalizations of Continuous Maps

Abstract

In this chapter we study different maps which generalize continuous maps. It will be shown by examples that in general all notions are not identical with continuous maps. However, under certain conditions some of them are coincident.

We shall particularly deal with almost continuous, nearly continuous, graphically continuous, approximately continuous, and semicontinuous maps, not necessarily in this order.

It is worth noting that this chapter considers many different concepts which have been labeled by the same name by different authors, thus causing confusion. Depending upon their usefulness in the wider sense, these notions have been renamed. Our arbitrary action here is solely designed to clear the confusion, and it is hoped that it will not fall short of this goal.

We have included the work of a number of mathematicians, although not each theorem has been labeled by the proper name, e.g., Husain [45–49], Smith [80], Stallings [82], Singal and Singal [77], and Long and McGehee [61], etc.