Almost irresolute functions
A subset s of a topological space \((X,\tau)\) is defined to be semi-open if there is an open set U such that \(U\subset S\subset cl U\), and semi- closed if its complement is semi-open. The semi-closure of S, denoted by scl S, is defined to be the intersection of all semi-closed sets containing S. A subset W is called a semi-neighbourhood of a point x of X if there is a semi-open set v such that \(x\in V\subset W\). \textit{K. K. Dube}, \textit{J. Lee} and \textit{O. S. Panwar} [UIT Rep. 14, 379-383 (1983; Zbl 0528.54011)] defined a function \(f: (X,\tau)\to (Y,\sigma)\) to be almost irresolute if for every point x of X and for each semi- neighbourhood V of f(x) in Y, scl \(f^{-1}(V)\) is a semi-neighbourhood of x in X. The author obtains several characterizations of almost irresolute functions and investigates their relationship to other weak forms of irresolute functions. In passing, the authors prove that four weak forms of irresolute fufnctions are in fact equivalent, namely the weak irresolute and \(\theta\)-irresolute functions of \textit{K. K. Dube}, \textit{G. Chae} and \textit{R. K. Sengar} [UIT Rep. 15, 373-379 (1984; Zbl 0586.54018)], the quasi irresolute functions of \textit{G. Di Maio} and \textit{T. Noiri} [Suppl. Rend. Circ. Mat. Palermo, II. Ser. 18, 255-273 (1988; Zbl 0663.54010)], and the almost irresolute functions of \textit{S. S. Thakur} and \textit{P. Paik} [J. Indian Acad. Math. 7, 63-66 (1985; Zbl 0592.54017)].
scientific article; zbMATH DE number 4105828