On \(\alpha\)-precontinuous functions (Q2706959)

Let \((X,\tau)\) be a topological space. A set \(S\) of \(X\) is said to be \(\alpha\)-open (resp. preopen, \(\beta\)-open) if \(S\subset \text{Int(Cl(Int} (S)))\) (resp. \(S\subset\text{Int(Cl}(S)), S\subset \text{Cl(Int(Cl} (S))))\). A function \(f:(X,\tau) \to(Y,\sigma)\) is said to be \(\alpha\)-continuous (resp. precontinuous, \(\beta\)-continuous) if \(f^{-1}(V)\) is \(\alpha\)-open (resp. preopen, \(\beta\)-open) for every open set \(V\) of \(Y\). A function \(f:(X,\tau) \to(Y, \sigma)\) is \(\alpha\)-irresolute (resp. almost \(\alpha\)-irresolute) if \(f^{-1}(V)\) is \(\alpha\)-open (resp. \(\beta\)-open) in \(X\) for every \(\alpha\)-open set \(V\) of \(Y\).NEWLINENEWLINENEWLINEIn the present paper, the authors introduce a new class of functions called \(\alpha\)-precontinuous functions. A function \(f:(X,\tau) \to (Y,\sigma)\) is said to be \(\alpha\)-precontinuous if \(f^{-1}(V)\) is preopen in \(X\) for every \(\alpha\)-open set \(V\) of \(Y\). Some characterizations and several fundamental properties of this class of functions are obtained. Relations between this type of functions and other classes of functions are obtained. The class of \(\alpha\)-precontinuous functions is contained in the class of almost \(\alpha\)-irresolute functions and contains the class of \(\alpha\)-irresolute functions.