Some types of convergence of sequences of functions

Abstract

Csaszar and Laczkovich defined and studied discrete convergence as well as the equal convergence for sequences of real-valued functions. Then definitions of uniformly equal convergence and uniformly discrete convergence were proposed by Papanastassiou as well. Nonetheless, the properties and relations of these convergent sequences of functions are not studied yet. In this project we will study the relations and the properties of the convergent sequences of functions. In addition, the properties of classes of functions which are uniformly equal limits and uniformly discrete limits will be then studied. The notion of - convergence for sequences of real-valued functions on a metric space will be introduced. The result is that if is a compact metric space then implies where denotes uniform convergence. Conversely, Hola and Salat showed that is true only if is a compact metric space, thus characterizing a compactness in terms of these convergences. Subsequently, we define new types of convergence called - uniform equal convergence. The relations of these convergences are then studied. The properties of these convergences are also investigated in order that we could obtain sufficient conditions for a metric space to be compact in term of it.