A note on Levine's decomposition of continuity
Levine's decomposition of continuity for an arbitrary function between arbitrary topological spaces is strengthened. A function \(f:X\to Y\) is weakly continuous if \(f^{-1}(V)\subset Int f^{-1}(Cl V)\) for each open set \(V\subset Y\) and f is \(w^*.c\). if \(f^{-1}(Bd V)\) is closed for each open \(V\subset Y\) where Int, Cl and Bd are the interior, closure and boundary operators respectively. Levine proved that a function is continuous if and only if it is both weakly continuous and \(w^*.c.\) A function \(f: X\to Y\) is weakly \(\alpha\)-continuous if and only if \(f^{-1}(V)\subset Int Cl Int f^{-1}(Cl V)\) for each open set \(V\subset Y\) and f is locally \(w^*.c\). if there is an open basis \({\mathcal B}\) for the topology on Y such that \(f^{-1}(Bd V)\) is closed for each \(V\in {\mathcal B}\). In this paper the following theorem is proved: A function \(f:X\to Y\) is continuous if and only if it is both weakly \(\alpha\)-continuous and locally \(w^*.c.\) Also some results in the literature are slightly improved.
scientific article; zbMATH DE number 4199005
scientific article