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Strong convergence and Dini theorems for non-uniform spaces. (English) Zbl 0888.54007
Summary: We use the strong convergence to obtain some new insight concerning the Dini theorem. A new concept generalizing the concepts of upper and lower semicontinuity is introduced and used to obtain some results for functions and multifunctions at the same time. A version of the Dini theorem for functions with non-uniform range space is proved, and some older results are improved. It is shown, that even classical assumptions of the Dini theorem imply a far stronger convergence than the mere uniform one.

MSC:
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54C08 Weak and generalized continuity
54C60 Set-valued maps in general topology
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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References:
[1] J.-P. Aubin and H. Frankowska , Set-valued Analysis , Birkhäuser , Boston - Basel - Berlin , 1990 . MR 1048347 | Zbl 0713.49021 · Zbl 0713.49021
[2] G. Beer , Topologies on closed and closed convex sets , Kluwner Academic press , 1994 . MR 1154502 | Zbl 0792.54008 · Zbl 0792.54008
[3] G. Beer , The approximation of upper semicontinuous multifunctions by step multifunctions , Pacific J. Math. 87 ( 1980 ), 11 - 19 . Article | MR 590862 | Zbl 0442.54014 · Zbl 0442.54014
[4] I. Del Prete , M.B. Lignola , Uniform convergence structures in function spaces and preservation of continuity , Ricerche di Matematica 36 fasc 1 ( 1987 ), 45 - 58 . MR 977052 | Zbl 0666.54011 · Zbl 0666.54011
[5] S. Francaviglia , A. Lechicki and S. Levi , Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions , J. Math. Anal. Appl . 112 ( 1985 ), 347 - 370 .. MR 813603 | Zbl 0587.54003 · Zbl 0587.54003
[6] I. Kupka , Banach fixed point theorem for topological spaces , Revista Colombiana de Matematicas XXVI ( 1992 ), 95 - 100 . Article | MR 1240486 | Zbl 0782.54039 · Zbl 0782.54039
[7] I. Kupka , Quasicontinuous selections for one-to-finite multifunctions , Acta Mathematica Universitatis Comenianae LVIII-LIX ( 1990 ), 207 - 214 . MR 1120366 | Zbl 0736.54011 · Zbl 0736.54011
[8] I. Kupka , Convergences preserving the fixed point property , Preprint ( 1997 ). MR 1857303 · Zbl 0985.54035
[9] I. Kupka and V. Toma , A uniform convergence for non-uniform spaces , Publicationes Mathematicae Debrecen 47 no. 3 -4 ( 1995 ), 299 - 309 . MR 1362292 | Zbl 0879.54003 · Zbl 0879.54003
[10] A. Lechicki , A generalized measure of noncompactness and Dini’s theorem for multifunctions , Rev. Roumaine Math. Pures Appl. 30 ( 1985 ), 221 - 227 . MR 789058 | Zbl 0584.54015 · Zbl 0584.54015
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