The notion of exhaustiveness and Ascoli-type theorems. (English) Zbl 1141.26001

Summary: 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 \(\alpha\)-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 \(\alpha\)-convergence. Finally we give conditions under which the pointwise limit of a sequence of arbitrary functions is a continuous function.


26A21 Classification of real functions; Baire classification of sets and functions
26A03 Foundations: limits and generalizations, elementary topology of the line
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