Gsα-OPEN SETS IN GRILL TOPOLOGICAL SPACES

Abstract

A non-null collection \(\mathcal{G}\) of subsets of a topological space \((X,\tau)\) is said to be a grill on \(X\) if (i) \(\emptyset \notin \mathcal{G}\), (ii) \(A \in \mathcal{G}\) and \(A \subseteq B\) implies that \(B \in \mathcal{G}\), (iii) \(A,B \subseteq X\) and \(A \cup B \in \mathcal{G}\) implies that \(A \in \mathcal{G}\) or \(B \in \mathcal{G}\). A triple \((X,\tau,\mathcal{G})\) is called a grill topological space. A new class \(\mathcal{G}_{s\alpha}O(X)\) of \(\mathcal{G}_{s\alpha}\)-open sets in a grill topological space with respect to \(\mathcal{G}\) is introduced. Also, we define \(\mathcal{G}_{s\alpha}\)-continuous and \(\mathcal{G}_{s\alpha}\)-open(closed) functions and study some of their important properties. In addition, we introduce a \((\mathcal{G},\mathcal{G}')_{s\alpha}\)-continuous function and investigate its properties.