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On \(m\)-continuous multifunctions. (English) Zbl 1049.54507

Summary: The authors define a multifunction \(F\colon X\to Y\) to be \(m\)-continuous if for each point \(x\in \) and any open subsets \(G_1,G_2\) of \(Y\) such that \(F(x)\subset G_1\) and \(F(x)\cap G_2\neq\emptyset\) there exists an \(m\)-open subset \(U\) of \(X\) containing \(x\) such that \(F(U)\subset\text{Cl}\,(G_1)\) and \(F(u)\cap\text{Cl}\,(G_2)\neq\emptyset\) for every \(u\in U\). The functions enable us to unify continuity, \(\alpha\)-continuity, quasi-continuity, precontinuity, and \(\beta\)-continuity for multifunctions. Some characterizations and several properties concerning \(m\)-continuous multifunctions are obtained.

MSC:

54C60 Set-valued maps in general topology
54C08 Weak and generalized continuity
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