全部 标题 作者 关键词 摘要

# Decomposition of Topologies Which Characterize the Upper and Lower Semicontinuous Limits of Functions

Abstract:

We present a decomposition of two topologies which characterize the upper and lower semicontinuity of the limit function to visualize their hidden and opposite roles with respect to the upper and lower semicontinuity and consequently the continuity of the limit. We show that (from the statistical point of view) there is an asymmetric role of the upper and lower decomposition of the pointwise convergence with respect to the upper and lower decomposition of the sticking convergence and the semicontinuity of the limit. This role is completely hidden if we use the whole pointwise convergence. Moreover, thanks to this mirror effect played by these decompositions, the statistical pointwise convergence of a sequence of continuous functions to a continuous function in one of the two symmetric topologies, which are the decomposition of the sticking topology, automatically ensures the convergence in the whole sticking topology. 1. Introduction Since the end of the nineteenth century several outstanding papers appeared to formulate a set of conditions, which are both necessary and sufficient, to be added to pointwise convergence of a sequence of continuous functions, to preserve continuity of the limit. Indeed, all classical kinds of convergences of sequences of functions between metric spaces (Dini, Arzelà, Alexandroff) are based on the pointwise convergence assumption that has been always considered a preliminary one. Recently, in [1, 2], Caserta et al. proposed a new model to investigate convergences in function spaces: the statistical one. Actually, they obtained results parallel to the classical ones, concerning the continuity of the limit, in spite of the fact that statistical convergence has a minor control of the whole set of functions. In  they proved that continuity of the limit of a sequence of functions is equivalent to several modes of statistical convergence which are similar, but weaker than the classical ones. A parallel to the classical results is expected since, after all, in  the authors found the statistical convergence to be the same as a very special regular triangular matrix summability method for bounded (and some unbounded) sequences. Thus many new results concerning statistical convergence follow from the corresponding known results for matrix summability. In 1969 Bouleau [4, 5] defined the sticking topology as the weakest topology finer than pointwise convergence to preserve continuity. In , Beer presented two new topologies on , finer than the topology of pointwise convergence, which are indeed the decomposition of the sticking

References

  A. Caserta and Lj. D. R. Ko？inac, “On statistical exhaustiveness,” . In press.  A. Caserta, G. Di Maio, and Lj. D. R. Ko？inac, “Statistical convergence in function spaces,” . In press.  M. K. Khan and C. Orhan, “Matrix characterization of A-statistical convergence,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 406–417, 2007.  N. Bouleau, “Une structure uniforme sur un espace F(E, F),” Cahiers de Topologie et Géométrie Différentielle Catégoriques, vol. 11, pp. 207–214, 1969.  N. Bouleau, “On the coarsest topology preserving continuity,” General Topology. In press, http://arxiv.org/abs/math/0610373v1.  G. Beer, “Semicontinuous limits of nets of continuous functions,” . In press.  W. J. Pervin, Foundations of General Topology, Academic Press, New York, NY, USA, 1964.  W. J. Pervin, “Quasi-uniformization of topological spaces,” Mathematische Annalen, vol. 147, pp. 316–317, 1962.  P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces, vol. 77 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1982.  J. L. Kelley, General Topology, D. Van Nostrand Company, Princeton, NJ, USA, 1955.  A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, UK, 2nd edition, 1979.  H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951.  R. C. Buck, “Generalized asymptotic density,” American Journal of Mathematics, vol. 75, pp. 335–346, 1953.  G. Di Maio and L. D. R. Ko？inac, “Statistical convergence in topology,” Topology and its Applications, vol. 156, no. 1, pp. 28–45, 2008.  J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985.  T. ？alát, “On statistically convergent sequences of real numbers,” Mathematica Slovaca, vol. 30, no. 2, pp. 139–150, 1980.  H. ？akalli and M. K. Khan, “Summability in topological spaces,” Applied Mathematics Letters, vol. 24, no. 3, pp. 348–352, 2011.  V. Gregoriades and N. Papanastassiou, “The notion of exhaustiveness and Ascoli-type theorems,” Topology and its Applications, vol. 155, no. 10, pp. 1111–1128, 2008.  G. Beer and S. Levi, “Strong uniform continuity,” Journal of Mathematical Analysis and Applications, vol. 350, no. 2, pp. 568–589, 2009.  A. Caserta, G. Di Maio, and L. Holá, “Arzelà's theorem and strong uniform convergence on bornologies,” Journal of Mathematical Analysis and Applications, vol. 371, no. 1, pp. 384–392, 2010.  P. S. Alexandroff, Einführung in die Mengenlehre und die Theorie der Reellen Funktionen, Deutscher Verlag der Wissenschaften, Berlin, Germany, 1956.  C. Arzelà, “Intorno alla continuà della somma di infinite funzioni continue,” Rend. R. Accad. d. Scienze dell'Istituto di Bologna, pp. 79–84, 1884.  R. G. Bartle, “On compactness in functional analysis,” Transactions of the American Mathematical Society, vol. 79, pp. 35–57, 1955.

Full-Text