Slightly \(\beta\)-continuous functions
A subset \(A\) of a topological space \((X,\tau)\) is said semi-open (resp. preopen, \(\beta\)-open) if \(A\subset \text{Cl(Int} (A))\) (resp. \(A\subset \text{Int(Cl}(A))\), \(A\subset\text{Cl(Int(Cl}(A))))\). A function \(f:(X,\tau) \to (Y, \sigma)\) is said to be slightly continuous (resp. slightly semi-continuous, faintly precontinuous) if for each \(x\in X\) and each open set \(V\) containing \(f(x)\) there exists an open (resp. semi-open set) \(U\) of \(X\) containing \(x\) such that \(f(U)\subset V\). In this paper the author introduces the notion of slightly \(\beta\)-continuous functions. A function \(f: (X,\tau) \to(Y,\sigma)\) is slightly \(\beta\)-continuous if for each \(x\in X\) and each clopen set \(V\) containing \(f(x)\) there exists a \(\beta\)-open set \(U\) containing \(x\) such that \(f(U)\subset V\). The notion of slightly \(\beta\)-continuity is implied by both slightly continuity and faint precontinuity. Some characterizations are obtained. Especially, the author defines the notion of ultra-regularization of a topology and obtains interesting characterizations of slightly \(\beta\)-continuous functions. Basic properties of these new types of functions are obtained. The relationships between slight \(\beta\)-continuity and several forms of \(\beta\)-continuity are investigated.
scientific article; zbMATH DE number 1780287