The notion of exhaustiveness and Ascoli-type theorems

In this paper we introduce the notion of exhaustiveness which applies for both families and nets of functions. This new notion is close to equicontinuity and describes the relation between pointwise convergence for functions and α-convergence (continuous convergence). Using these results we obtain some Ascoli-type theorems dealing with exhaustiveness instead of equicontinuity. Also we deal with the corresponding notions of separate exhaustiveness and separate α-convergence. Finally we give conditions under which the pointwise limit of a sequence of arbitrary functions is a continuous function.