On some forms of quasi-uniform convergence of transfinite sequence of multifunctions

• Autorzy: Przemski M.
• Języki publikacji: EN

Abstrakty

EN

In this paper we introduce various forms of convergence of transfinite sequences of multifunctions with values in a quasi-uniform space. We also study some weak types of continuity for such multifunctions. Any such sequence of multifunctions generates the sequence of the sets of weak types of continuity points and the sequence of various types of cluster sets of members of such sequence. We study the connection between convergence of a transfinite sequences of multifunctions and convergence of the corresponding sequences of the sets of the weak continuity points and the sequences of cluster sets. Some of the presented results concern of general nets of multifunctions.

Słowa kluczowe

EN

convergence; quasiuniform; multifunctions; generalized continuity; hyperspace; cluster sets

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2010
• Tom: Vol. 50, [Z] 1
• Strony: 3--21
• Opis fizyczny: Bibliogr. 42 poz.

Twórcy

autor

Przemski M.

• Department of Applied Mathematics, Warsaw University of Life Sciences ul. Nowoursynowska 159 bud. 34, 02-776 Warsaw, Poland,

Bibliografia

• [1] D. Andrijevicc, Some properties of the topology of α-open sets, Mat. Vesnik 36 (1984), 1-10.
• [2] G. Aumann, Reelle Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 68, Springer-Verlag, Berlin, Gottingen, Heidelberg, 1954.
• [3] G. Beer, Topologies on closed and closed convex sets, Kluwer Academic Publishers, Dortdrecht, Holland, 1993.
• [4] G. Berthiaume, On quasi-uniformities in hyperspaces, Proc. Amer. Math. Soc. 66 (1977), 335-343.
• [5] J. Cao, Quasiuniform convergence for multifunctions, J. Math. Research & Exposition 14 (1994), 327-331.
• [6] J. Cao, I.L. Reilly, Almost Quasiuniform convergence for multifunctions, Comment. Math. 36 (1996), 57-68.
• [7] J. Cao, I.L. Reilly, M.K. Vamanamurthy, Comparison of convergences for multifunctions, Demonstratio Math. Vol.XXX, No 1 (1997), 172-182.
• [8] J. Ewert, On the quasi-uniform convergence of transfinite sequence of functions, Acta Math. Univ. Comenianae Vol. LXII, 2 (1993), 221-227.
• [9] P. Fletcher, W.F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York, 1982.
• [10] S. Francaviglia, A. Lechicki, S.Levi, Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), 347-370.
• [11] R. Hrycay, Noncontinuous multifunctions, Pacific F. Math. 35 (1970), 141-154.
• [12] J. Kelley, General topology, Van Nostrand, New York,1955.
• [13] S. Kempisty, Sur les functions quasicontinues, Fund. Math. 19 (1932), 184-197.
• [14] E. Klein, A.C. Thompson, Theory of Correspondences, John Wiley & Sons, 1984.
• [15] K. Kuratowski, Topology, Vol.I, New York, 1966.
• [16] H.P.A. Künzi, Nonsymmetric topology, Bolyai Soc. Math. Stud., Topology, Szeksard, Hungary (Budapest), 4 (1995), 303-338.
• [17] H.P.A. Künzi, C. Ryser, The Bourbaki quasi-uniformity, Topology Proc., 20 (1995), 161-183.
• [18] M. Lassonde, J. Revalski, Fragmentability of sequenses of set-valued mappings with applications to variational principles, Proc. Amer. Math. Soc. 133 (2005), 2637-2646.
• [19] A. Lechicki, S. Levi, A. Spakowski, Bornological convergence, J. Math. Anal. Appl. 297 (2004), 751-770.
• [20] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36-41.
• [21] N. Levine, W.J. Stager, On the hyperspaces of a quasi-uniform space, Proc. Amer. Math. Soc., 15 (1971), 101-106.
• [22] M. Matejdes, Minimal multifunctions and the cluster sets, Tatra Mt. Math. Pub. 34 (2006), 71-76.
• [23] A.S. Mashhour, I.A. Hasanein, S.N. El-Deeb, α-continuous and _-open mappings, Acta Math. Hungar., 41 (1983), 213-218.
• [24] A.S. Mashhour, M.E. Abd El-Monsef, S.N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47-53.
• [25] E. Michael, Topologies on spaces of sets, Trans. Amer. Math. Soc. 71 (1951), 152-182.
• [26] M.E. Abd El-Monsef, S.N. El-Deeb, R.A. Mahmoud, α-open sets and αcontinuous mappings, Bull. Fac. Sci. Assiut Univ., 12 (1983), 77-90.
• [27] W.B. Moors, J.R. Giles, Generic continuity of minimal set-valued mappings, J. Austral. Math. Soc. Ser. A 63 (1997), no.2, 238-262.
• [28] S. Mrówka, On the convergence of net of sets, Fund. Math. 45 (1958), 237-246.
• [29] M.G. Murdeshwar, S.A. Naimpally, Quasi-uniform topological spaces, P. Noordhoff Ltd., Rroningen, 1966.
• [30] T. Neubrunn, Strongly quasi-continuous multivalued mappings, General Topology and its Relation to Modern Analisis and Algebra VI (Prague 1986) Heldermann, Berlin, 1988, 351-359.
• [31] O. Njastad, On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961-970.
• [32] W.J. Pervin, Quasi-uniformization of topological spaces, Math. Ann. 147 (1962), 316-317.
• [33] V.I. Ponomarev, Properties of topological spaces preserved under multivalued continuous mappings on compacta, Amer. Math. Soc. Trans. 38 (2)(1964), 119-140.
• [34] V. Popa, Some properties of H-almost continuous multifunctions, Problemy Mat. 10 (1990), 9-26.
• [35] V. Popa, On a decomposition of quasicontinuity for multifunctions, Stud. Cerc. Mat. 27 (1975), 323-328.
• [36] V. Popa, T. Noiri, On upper and lower β-continuous multifunctions, Real Analysis Exchange 22 (1996/97), 362-367.
• [37] M. Predoi, Sur la convergence quasi-uniforme, Periodica Math. Hungar. 10 (1979), 31-40.
• [38] M. Przemski, On the relationships between the graphs of multifunctions and some forms of continuity, Demonstratio Math. XLI (2008), 203-224.
• [39] M. Przemski, Cluster sets and related properties of multifunctions, Demonstratio Math. XLII, No 1 (2009), 203-217.
• [40] Ch. Richter, I. Stephani, Cluster sets and approximation properties of quasi-continuous and cliquish functions, Real Anal. Exchange 29 (2003-4), 229-322.
• [41] A. Spakowski, On lower semicontinuous multifunctions in quasi-uniform and vector spaces, Proceedings of the Ninth Prague Topological Symposium (Prague 2001), Topology Atlas. Toronto 2002, 309-319.
• [42] J.D. Weston, Some theorems on cluster sets, J. London Math. Soc., 33 (1958), 458-441.