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

被引:0
作者
Caserta, Agata [1 ]
机构
[1] SUN, Dept Math, I-81100 Caserta, Italy
关键词
STATISTICAL CONVERGENCE; SPACES;
D O I
10.1155/2011/857278
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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.
引用
收藏
页数:9
相关论文
共 23 条
  • [1] ALEXANDROFF PS, 1956, EINFUHRUNG MENGENLEH
  • [2] Arzela C., 1884, REND R ACCAD SCI I B, P79
  • [3] Bartle R.G., 1955, T AM MATH SOC, V79, P35, DOI DOI 10.1090/S0002-9947-1955-0073938-2
  • [4] Beer G., SEMICONTINU IN PRESS
  • [5] Strong uniform continuity
    Beer, Gerald
    Levi, Sandro
    [J]. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2009, 350 (02) : 568 - 589
  • [6] Bouleau N., 1969, CAH TOPOL GEOM DIFF, V11, P207
  • [7] Bouleau N., GEN TOPOLOG IN PRESS
  • [8] GENERALIZED ASYMPTOTIC DENSITY
    BUCK, RC
    [J]. AMERICAN JOURNAL OF MATHEMATICS, 1953, 75 (02) : 335 - 346
  • [9] Summability in topological spaces
    Cakalli, H.
    Khan, M. K.
    [J]. APPLIED MATHEMATICS LETTERS, 2011, 24 (03) : 348 - 352
  • [10] Caserta A., STAT EXHAUS IN PRESS