: In this paper, some characterizations and proportion of notion a investigated. Throughout this paper (X, ?) and (Y, ? ) (simply, X and Y) represent topological spaces on which separation axioms are assumed unless otherwise mentioned. We introduce a new class of sets called regular generalized open sets which is properly placed in between the class of open sets and the class of - open sets. Throughout this paper (X, ?) represents a topological space on which no separation axiom is assumed unless otherwise mentioned. For a subset A of a topological space X, cl (A) and int (A) denote the closure of A and the interior of A respectively. X/A or Ac denotes the complement of A in X. introduced and investigated semi open sets, generalized closed sets, regular semi open sets, weakly closed sets, semi generalized closed sets , weakly generalized closed sets, strongly generalized closed sets, generalized pre - regular closed sets, regular generalized closed sets, and generalized ?-generalized closed sets respectively.
A topological space (X, ?) is said to be ?*-connected if it is not the union of two nonempty disjoint ?*- open sets. If (X, ?) is a ?* - connected space and f:(X, ?) ? (Y, ?) has a (?*, ?)-graph and ?*- continuous function, the constant. Suppose that f is not constant. There exist disjoint points x, y ? X such that f(x) = f(y). Since (x, f(x)) G (f), then y ? f(x), hence by, there exist open sets U and V containing x and f(x) respectively such that f(U) ? V = ?. Since f is ? * - continuous, there exist a ? * - open sets G containing y such that f(G) ? V. Since U and V are disjoint ?*- open sets of (X, ?), it follows that (X, ?) is not ?*- connected Therefore, f is constant. Let (X1, ?1), (X2, ?2) and (X, ?) be topological spaces. Define a function f: (X, ?) ? (X1 × X2, ?1 × ?2) by f(x) = (f(x1), f(x2)). Then fi: X ? (Xi, ?i), where (i = 1, 2) is ?*-continuous if f is ?*- continuous.
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