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Classification of points of lower semi-continuity of a multifunction in topological spaces. (English) Zbl 1244.54033

Summary: We introduce the notion of \(y\)-lower semi-continuity and point out a distinction between a point of lower semi-continuity in the global sense and a point of lower semi-continuity in the local sense in general topological spaces after classifying points of \(y\)-lower semi-continuity (resp. lower semi-continuity) and also study their interrelationships. In particular, we find a necessary and sufficient condition for a bijective open multifunction on a \(T_2\) space to be lower semi-continuous. Finally, a sufficient condition for an open bijective multifunction on the real line to have at most countably many points of lower semi-discontinuity is formulated.

MSC:

54C08 Weak and generalized continuity
54C05 Continuous maps
54C60 Set-valued maps in general topology
54A99 Generalities in topology
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