Another weak form of faint continuity
Let \((X,\tau)\) be a topological space. A set \(A\) of \(X\) is said to be \(\alpha\)-open [\textit{O. Njåstad}, Pac. J. Math. 15, 961-970 (1965; Zbl 0137.41903)] (resp. \(b\)-open [\textit{D. Andrijević}, Mat. Vesn. 48, No. 1-2, 59-64 (1996; Zbl 0885.54002)] or sp-open [\textit{J. Dontchev}, and \textit{M. Przemski}, Acta Math. Hung. 71, No. 1-2, 109-120 (1996; Zbl 0852.54012)] or \(\gamma\)-open [\textit{A. A. El-Atik}, A study of some types of mappings on topological spaces, MSc-Thesis, Tanta Univ., Egypt (1997)]) if \(A \subset\text{Int(Cl(Int}(A)))\), resp. \(A\subset\text{Cl(Int}(A))\cup \text{Int(Cl}(A))\). A subset \(A\) is said to be \(\theta\)-open [\textit{P. E. Long} and \textit{L. L. Herrington}, Kyungpook Math. J. 22, 7-14 (1982; Zbl 0486.54009)] if for each \(x\in A\) there exists an open set \(U\) such that \(x\in U\subset\text{Cl}(U)\subset A\). Long and Herrington defined a weak form of continuity by making use of \(\theta\)-open sets. \textit{T. Noiri} and \textit{V. Popa} [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 34(82), No. 3, 263-270 (1990; Zbl 0739.54003)] introduced and investigated three weaker forms of faint continuity which are called faint semicontinuity, faint precontinuity and faint \(\beta\)-continuity. In this paper the author introduces and investigates other weak forms of faint continuity, namely faint \(\alpha\)-continuity and faint \(\gamma\)-continuity. A function \(f:(X,\tau)\to (Y,\sigma)\) is said to be faintly \(\alpha\)-continuous (resp. faintly \(\gamma\)-continuous) if for each \(x\in X\) and each \(\theta\)-open set \(V\) of \(Y\) containing \(f(x)\) there exists an \(\alpha\)-open (resp. \(\gamma\)-open) set \(U\) containing \(x\) such that \(f(U)\subset V\). Some characterizations and basic properties of these new types of functions are obtained. The relationships between these functions and several weak forms of faint continuity are investigated. Finally, the author mentions that the present paper may be relevant to fractal and Cantorian physics.
scientific article; zbMATH DE number 1755328
scientific article