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Generalized statistical convergence and convergence free spaces. (English) Zbl 0955.40001

The statistical convergence was introduced by H. Fast in 1951. Since then this notion has been studied by many authors including J. Connor, J. A. Fridy, H. I. Miller, A. R. Freedman, J. J. Sember, I. J. Maddox, T. Salat, etc. In this paper, by using the concepts of convergence free spaces, cofilters and filters based on a countably infinite set, the authors introduce and study a generalized statistical convergence. Algebraic and order properties, preservation under uniform convergence, Cauchy properties, and properties of cluster points are derived. A relationship between this generalized statistical convergence and subsets of the Stone-Čech compactification of integers is pointed out.

MSC:

40A05 Convergence and divergence of series and sequences
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
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