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On statistical exhaustiveness. (English) Zbl 1255.54010

The authors study statistical versions of several types of convergence of sequences of functions between two metric spaces. They introduce the notion of statistical exhaustiveness of a sequence of functions between two metric spaces and establish a characterisation of this notion using statistical density. The authors define statistical \(\alpha\)-convergence of a sequence of functions in \(Y^X\) (\(X,Y\) being two metric spaces) and establish its equivalence with other notions of convergence (under some additional hypothesis) viz. statistical point-wise convergence, statistical uniform convergence on compacta and statistical strong uniform convergence on the bornology \(\mathcal K_r\) (the collection of all nonempty relatively compact subsets of \(X\) i.e. subsets of \(X\) with compact closure); a bornology on \(X\) is a family \(\mathcal B\) of nonempty subsets of \(X\) which is closed under finite unions, is hereditary (i.e. closed under taking nonempty subsets) and forms a cover of \(X\). The authors introduce the notion of statistical weak exhaustiveness of a sequence of functions in \(Y^X\) and show that for a sequence \((f_n)_{n\in\mathbb N}\) \big[in \(C(X,Y)\)\big] which is statistically point-wise convergent to \(f\in Y^X\), the statistical weak exhaustiveness of \((f_n)\) is equivalent to statistical strong uniform convergence of \((f_n)\) to \(f\) on the bornology \(\mathcal F\) (the family of all nonempty finite subsets of \(X\)) and is also equivalent to the continuity of \(f\). Finally the authors prove that if a sequence \((f_n)_{n\in\mathbb N}\) in \(C(X,Y)\) is statistically Alexandroff convergent to \(f\in Y^X\) then \(f\) is continuous.

MSC:

54C35 Function spaces in general topology
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
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[1] Engelking, R., General Topology (1989), Heldermann Verlag: Heldermann Verlag Berlin · Zbl 0684.54001
[2] Kelley, J. L., General Topology (1955), D. Van Nostrand Company, Inc.: D. Van Nostrand Company, Inc. Princeton · Zbl 0066.16604
[3] Beer, G.; Levi, S., Strong uniform continuity, J. Math. Anal. Appl., 350, 568-589 (2009) · Zbl 1161.54003
[4] Gregoriades, V.; Papanastassiou, N., The notion of exhaustiveness and Ascoli-type theorems, Topology Appl., 155, 1111-1128 (2008) · Zbl 1141.26001
[5] Zygmund, A., Trigonometric Series (1979), Cambridge University Press: Cambridge University Press Cambridge
[6] Fast, H., Sur la convergence statistique, Colloq. Math., 2, 241-244 (1951) · Zbl 0044.33605
[7] Steinhaus, H., Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2, 73-74 (1951)
[8] Di Maio, G.; Kočinac, Lj. D.R., Statistical convergence in topology, Topology Appl., 156, 28-45 (2008) · Zbl 1155.54004
[9] Fridy, J. A., On statistical convergence, Analysis, 5, 301-313 (1985) · Zbl 0588.40001
[10] Šalát, T., On statistically convergent sequences of real numbers, Math. Slovaca, 30, 139-150 (1980) · Zbl 0437.40003
[11] Hu, S.-T., Boundedness in a topological space, J. Math. Pures Appl., 28, 287-320 (1949) · Zbl 0041.31602
[12] Hogbe-Nlend, H., Bornologies and Functional Analysis (1977), North-Holland: North-Holland Amsterdam · Zbl 0359.46004
[13] Caserta, A.; Di Maio, G.; Holá, L’., Arzelà’s theorem and strong uniform convergence on bornologies, J. Math. Anal. Appl., 371, 384-392 (2010) · Zbl 1202.54004
[14] A. Caserta, G. Di Maio, Lj.D.R. Kočinac, Bornologies, selection principles and function spaces, Topology Appl. (in press).; A. Caserta, G. Di Maio, Lj.D.R. Kočinac, Bornologies, selection principles and function spaces, Topology Appl. (in press). · Zbl 1253.54021
[15] Alexandroff, P. S., Einführung in die Mengenlehre und die Theorie der reellen Funktionen (1956), Deutscher Verlag der Wissenschaften, Translated from the 1948 Russian edition · Zbl 0070.04704
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