On statistical exhaustiveness. (English) Zbl 1255.54010

Caserta, A.; Kočinac, Lj. D. R.

Appl. Math. Lett. 25, No. 10, 1447-1451 (2012).

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.

Reviewer: Sandip Jana (Kolkata)

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)

Keywords:

statistical convergence; statistical uniform convergence on compacta; bornology; (weak) exhaustiveness; statistical strong uniform convergence on a bornology

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