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Strongly quasi-continuous multivalued mappings. (English) Zbl 0655.54014

General topology and its relations to modern analysis and algebra VI, Proc. 6th Symp., Prague/Czech. 1986, Res. Expo. Math. 16, 351-359 (1988).
[For the entire collection see Zbl 0632.00016.]
Let X and Y be topological spaces. A multifunction F: \(X\to Y\) is said to be \(\alpha\)-lower-continuous if \(F^-(V)\subset int cl int F^-(V)\) for every open \(V\subset Y\); it is upper-almost-continuous if \(F^+(V)\subset int cl F^+(V)\) for every open \(V\subset Y\). The author proves that if a multifunction F: \(X\to Y\), where Y is a normal space, is closed-valued, \(\alpha\)-lower-continuous and upper-almost-continuous, then it is continuous. Other sufficient conditions for continuity of \(\alpha\)-continuous multifunctions are also given.
Reviewer: K.Nikodem

MSC:

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

Citations:

Zbl 0632.00016