On upper and lower \(D^*\)-supercontinuous multifunctions (Q2460664)

Let \((X,\tau)\) be a topological space. A subset \(F\) of \((X,\tau)\) is said to be strongly \(F_\tau\)-open if there exists a countable open complementary system \(\beta(F)\) with \(F\in \beta(F)\). In this paper the notions of upper/lower \(D^*\)-supercontinuous multifunctions are introduced: Definition. A multifunction \(F:X\to Y\) is said to be: a) upper \(D^*\)-supercontinuous at a point \(x\in X\) if for every open set \(V\) with \(F(x)\subset V\), there exists a strongly open \(F_\tau\)-set \(U\) containing \(x\) such that \(F(U)\subset V\); b) lower \(D^*\)-supercontinuous at a point \(x\in X\) if for every open set \(V\) with \(F(x)\cap V\neq \emptyset\), there exists a strongly open \(F_\tau\)-set \(U\) containing \(x\) such that \(F(u)\cap V\neq \emptyset\) for every \(u\in U\). Some characterizations and several properties of upper/lower \(D^*\)-supercontinuous multifunctions are obtained.