ABOUT VARIATIES OF G-SEQUENTAILLY METHODS, G-HULLS AND G-CLOSURES

Abstract

In the first countable spaces many topological concepts such as open and closed subsets; and continuous functions are defined via convergent sequences. The concept of limit defines a function from the set of all convergence sequences in X to X itself if X is a Hausdorff space. This is extended not only to topological spaces but also to sets. More specifically a G-method is defined to be a function defined on a subset of all sequences (see [7] and [10]). We say that a sequence x = (x_n) G-convergences to a if G(x) = a. Then many topological objects such as open and closed subsets and many others including these sets have been extended in terms of G-convergence. G-continuity, G-compactness and G connectedness have been studied by several authors ([1], [2], [3], [4]). On the other hand we know that in a topological space X, a sequence (x_n) converges to a point a ∈ X if any open neighbourhood of a includes all terms except finite number of the sequence. Similarly we define a sequence (x_n) to be G-sequentially converging to a if any G-open neighbourhood of a includes almost all terms. In this work provided some examples we indicate that G-convergence and G-sequentially convergence are different. We will prove that G-closed and G-sequentially closedness of subsets and therefore many others are different.