On rarely precontinuous functions (Q2706960)

Let \((X,\tau)\) be a topological space. A set \(S\) of \(X\) is said to be preopen if \(S\subset\text{Int(Cl}(S))\). A function \(f:(X,\tau) \to(Y, \sigma)\) is precontinuous [\textit{A. S. Mashhour}, \textit{M. E. Abd El-Monsef}, and \textit{S. N. El-Deep}, Proc. Math. Phys. Soc. Egypt 53, 47-53 (1982; Zbl 0571.54011)] if \(f^{-1}(G)\) is preopen for every open set \(G\) of \(Y\). A function \(f\) is almost weakly continuous [\textit{D. S. Janković}, Int. J. Math. Math. Sci. 8, 615-619 (1985; Zbl 0577.54012)] if \(f^{-1}(G)\subset \text{Int(Cl}(f^{-1} (\text{Cl}(G))))\) for every open set \(G\) of \(Y\).NEWLINENEWLINENEWLINEIn this paper, the author introduces a new class of functions called rarely precontinuous functions. A function \(f\) is rarely continuous [\textit{V. Popa}, Glas. Mat., III. Ser. 14(34), 359-362 (1979; Zbl 0451.54012)] (resp. rarely precontinuous) if for each \(x\in X\) and each open set \(G\) of \(Y\) containing \(f(x)\) there exist a rare set \(R_G\) with \(G\cap\text{Cl}(R_G)= \emptyset\) and an open (resp. preopen) set \(U\) containing \(x\) such that \(f(U) \subset G\cup R_G\). The class of rarely precontinuous functions is a generalization of both the classes of rare continuous functions and that of almost weakly continuous functions. Some characterizations and several fundamental properties of this class of functions are obtained. In the last part a decomposition of precontinuity is obtained.