Abstract
Using the notions of local uniform and strong local uniform con-vergence for the sequence of real valued functions or with value in metric space, the class of locally equally and strong locally equally convergences are studied. We are concern to dependence of type of some convergences from the neighborhood of the limit point. The known locally uniformly convergence is a key of some applications of this idea. We can reformulate one type of Arzela Theorem and nd relations of this convergence with quasi-uniformly by segments of Alexandro off convergence. Beside this type of convergence, we focus to another convergence which is nearer the well known a-convergence.
Highlights
There are many authors that try to solve some rebuses of relations of pointwise convergence and other convergences
Using the notion of the exhaustive family given from Gregoriades and Papanastassiou [2] and the notion of α-convergence has been known by the beginning of the 20th century from work of Stoilov [12] and Arens [6] in 1950s, in the third section, we have present the notion of strong uniformly locally convergence which is stronger α-convergence but weaker uniform convergence
By means of uniformly local convergence in real valued functions, in the section fourth, we give any variant of the type of Arzela Theorem and one equivalent claim in the Alexandroff Theorem
Summary
There are many authors that try to solve some rebuses of relations of pointwise convergence and other convergences. One of this problems is: what conditions we must be added to point-wise convergence of continuous functions to preserve continuity? Das and Papanesstassiu [3] give some very interesting convergences as and αequally convergences, follow them we focus in some similar propositions for the uniform locally an strong uniformly locally sequences
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