On Various Modes of Convergence and Notions of Exhaustiveness With Korovkin‐Type Theorems

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Abstract
In this paper, we introduce refined notions related to convergence and exhaustiveness for sequences of functions defined between metric spaces. These include rigid uniform alpha convergence as a strengthened variant of alpha convergence, along with uniform sequential exhaustiveness, rigid uniform exhaustiveness, Cauchy exhaustiveness, and rigid Cauchy exhaustiveness as enhanced forms of exhaustiveness. Furthermore, using these new concepts, we investigate their properties and relationships among these different notions. Lastly, we give Korovkin‐type theorems for certain notions of exhaustiveness; it will be observed that the conditions in the classical Korovkin theorem can be weakened.

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