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Some topological properties of the set of filter cluster functions. (English) Zbl 1463.40019

Summary: In [the first and second author, Filomat 27, No. 8, 1373–1383 (2013; Zbl 1324.40007)], we generalized the concepts of pointwise convergence, uniform convergence and \(\alpha\)-convergence for sequences of functions on metric spaces by using the filters on \(\mathbb N\). In this work, we define the concepts of limit function, \(\mathcal{F}\)-limit function and \(\mathcal{F}\)-cluster function respectively for each of these three types of convergence, where \(\mathcal{F}\) is a filter on \(\mathbb N\). We investigate some topological properties of the sets of \(\mathcal{F}\)-pointwise cluster functions, \(\mathcal{F}\)-\(\alpha\)-cluster functions and \(\mathcal{F}\)-uniform cluster functions by using pointwise and uniform convergence topologies.

MSC:

40A35 Ideal and statistical convergence
40A30 Convergence and divergence of series and sequences of functions
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)

Citations:

Zbl 1324.40007
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