Weak and strong forms of \(\beta\)-irresolute functions (Q1878602)

Let \(X\), \(Y\) be topological spaces. A function \(f:X\to Y\) is said to be weakly (resp. strongly) \(\beta\)-irresolute if for each point \(x\in X\) and each semi-preopen subset \(V\subset Y\) containing \(f(x)\), there exists a semi-preopen subset \(U\subset X\) containing \(x\) such that \(f(U)\subset V\cup\text{Int}\, Cl\,\text{Int}\, V\) (resp. \(f(U\cup\text{Int}\, Cl\,\text{Int}\, U)\subset V\)). A subset \(A\) is called semi-preopen if \(A\subset Cl\,\text{Int}\, Cl\, A\). The paper contains various characterizations of the above substitute of continuity and comparisons with other substitutes of similar kind. Semipre-regular spaces are introduced as a natural domain for such functions.