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Quasi-continuity of multivalued maps with respect to the qualitative topology. (English) Zbl 0724.54016
A multivalued map F:X\(\to Y\), where X,Y are topological spaces, is called upper (lower) quasicontinuous at \(x_ 0\in X\) if for each open \(V\subset Y\) such that \(F(x_ 0)\subset V\) \((F(x_ 0)\cap V\neq \emptyset)\) and for each neighbourhood U of \(x_ 0\) there exists an open set \(U_ 1\subset U\) such that F(x)\(\subset V\) (F(x)\(\cap V\neq \emptyset)\) for all \(x\in U_ 1\). The author investigates the quasicontinuity of multivalued maps with respect to the qualitative topology on Y (i.e. the topology \(T_ q=\{U\setminus H:U\) is open, H is of the first category\(\}\).
MSC:
54C60 Set-valued maps in general topology
54C08 Weak and generalized continuity
54E52 Baire category, Baire spaces
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