Generalized continuity and uniform approximation by step functions (Q2497013)

Let \((X,\tau)\) be a topological space. A subset \(A\) of \(X\) is called semi-open (resp. preopen, or nearly open, \(\alpha\)-open, \(\beta\)-open or semi-preopen) if \(A\subset Cl(Int(A))\) (resp., \(A\subset Int(Cl(A))\), \(A\subset Int(CL(Int(A)))\). A function \(f:(X,\tau)\to(Y,\sigma)\) is said to be quasi-continuous (resp. precontinuous or nearly continuous, \(\alpha\)-continuous, \(\beta\)-continuous or almost quasicontinuous) if \(f^{-1}(V)\) is semi-open (resp. preopen, \(\alpha\)-open, \(\beta\)-open) for every open set \(V\) of \(Y\). A function \(f:(X,\tau)\to (Y,\sigma)\) is said to be somewhat continuous if \(Int(f^{-1}(V))\neq\emptyset\) for every open set \(V\) of \(Y\) with \(f^{-1}(V)\neq\emptyset\). In this paper the author introduces the notion of \({\mathcal O}_*\)-continuity in this way: Let \((X,\tau)\) be a topological space and \({\mathcal O}_*\) a family of the power set \({\mathcal P}(X)\). A function \(f:(X,\tau)\to(Y,\sigma)\) is called \({\mathcal O}_*\)-continuous if \(f^{-1} (V)\in {\mathcal O}_*\) for every open set \(V\) of \(Y\). Clearly, quasi-continuous, precontinuous, \(\alpha\)-continuous, \(\beta\)-continuous, and somewhat continuous functions are particular cases of \({\mathcal O}_*\)-continuous functions. A function \(\varphi:(X,\tau)\to(Y,\sigma)\) is called an \({\mathcal O}_*\)-step function if there exists a partition \({\mathcal P}\) of \(X\) into \({\mathcal O}_*\)-open sets such that \(\varphi\) is constant on every member of \({\mathcal P}\). Every \({\mathcal O}_*\)-step function is \({\mathcal O}_*\)-continuous. In Theorems 1 and 2 by [\textit{C. Richter} and \textit{I. Stephani}, Real Anal. Exch. 29, No. 1, 99--322 (2003--2004; Zbl 1068.54015)] the author characterized quasicontinuous functions as uniform limits of semi-open step functions. In the present paper, the author gives representations of \({\mathcal O}_*\)-continuous functions as uniform limits of \({\mathcal O}_*\)-step functions. In particular the cases of precontinuous, \(\alpha\)-continuous, \(\beta\)-continuous and somewhat continuous functions are studied.