Abstract: A new class of sets called β-generalized closed sets and β-generalized open sets in topological spaces and its properties are studied. A subset A of a topological spaces (X, τ) is called β-generalized closed sets (briefly βg-closed) if cl(int(cl(A))) contains U whenever A contains U and U is open in X. A new class of βg-continuous maps and βg-irresolute maps in topological spaces and study some of its basic properties. In this study, we introduce the notion of βg (θ) convergence and βg (θ)-adherence in grill topological spaces and study some of its basic properties and relations among them.