On upper and lower \(\beta\)-continuous multifunctions
Let \(X,Y\) be topological spaces and \(F\) be a multifunction from \(X\) into \(Y\). We say that \(F\) is upper (lower) \(\beta\)-continuous if for each \(x\in X\) and each open set \(V\) containing \(F(x)\) \((F(x) \cap V \not = \emptyset)\) there exists \(U\in \beta (X,x)\) such that \(F(U) \subseteq V\) \((U\subseteq \{t\in X:F(t) \cap V \not = \emptyset \})\). Here, \(\beta (X,x)\) denotes the system of all \(\beta\)-open sets of \(X\) containing \(x\), and a subset \(A\) of \(X\) is said to be \(\beta\)-open if \(A\subseteq \operatorname {Cl}(\operatorname {Int}(\operatorname {Cl}(A)))\). This article contains some characterizations of upper (lower) \(\beta\)-continuous multifunctions, also for ``local'' case (upper (lower) \(\beta\)-continuity at a point), the relationships between them and quasi-continuous multifunctions and their properties.
scientific article; zbMATH DE number 1095904