On I-quotient mappings and I-cs'-networks under a maximal ideal

[EN] Let I be an ideal on N and f : X → Y be a mapping. f is said to be an I-quotient mapping provided f−1(U) is I-open in X, then U is I-open in Y . P is called an I-cs′-network of X if whenever {xn}n∈N is a sequence I-converging to a point x ∈ U with U open in X, then there is P ∈ P and some n0 ∈ N such that {x, xn0} ⊆ P ⊆ U. In this paper, we introduce the concepts of I-quotient mappings and I-cs′-networks, and study some characterizations of I-quotient mappings and I-cs′- networks, especially J -quotient mappings and J -cs′-networks under a maximal ideal J of N. With those concepts, we obtain that if X is an J -FU space with a point-countable J -cs′-network, then X is a meta-Lindelöf space.Zhou, X. (2020). On I-quotient mappings and I-cs'-networks under a maximal ideal. Applied General Topology. 21(2):235-246. https://doi.org/10.4995/agt.2020.12967OJS235246212J. R. Boone and F. Siwiec, Sequentially quotient mappings, Czech. Math. J. 26 (1976), 174-182.L. X. Cheng, G. C. Lin, Y. Y. Lan and H. Liu, Measure theory of statistical convergence, Sci. China Ser. A 51 (2008), 2285-2303. https://doi.org/10.1007/s11425-008-0017-zL. X. Cheng, G. C. Lin and H. H. Shi, On real-valued measures of statistical type and their applications to statistical convergence, Math. Comput. Modelling 50 (2009), 116-122. https://doi.org/10.1016/j.mcm.2009.04.004P. Das, Some further results on ideal convergence in topological spaces, Topol. Appl. 159 (2012), 2621-2626. https://doi.org/10.1016/j.topol.2012.04.007P. Das and S. Ghosal, When I-Cauchy nets in complete uniform spaces are I-convergent, Topol. Appl. 158 (2011), 1529-1533. https://doi.org/10.1016/j.topol.2011.05.006P. Das, Lj.D.R. Kocinac and D. Chandra, Some remarks on open covers and selection principles using ideals, Topol. Appl. 202 (2016), 183-193. https://doi.org/10.1016/j.topol.2016.01.003G. Di Maio and Lj. D. R. Kocinac, Statistical convergence in topology, Topol. Appl. 156 (2008), 28-45. https://doi.org/10.1016/j.topol.2008.01.015R. Engelking, General Topology (revised and completed edition), Heldermann Verlag, Berlin, 1989.H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244. https://doi.org/10.4064/cm-2-3-4-241-244L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, Princeton, 1960. https://doi.org/10.1007/978-1-4615-7819-2P. Kostyrko, T. Salát and W. Wilczynski, I-convergence, Real Anal. Exch. 26 (2000/2001), 669-686. https://doi.org/10.2307/44154069B. K. Lahiri and P. Das, I and I*-convergence in topological spaces, Math. Bohemica 130, no. 2 (2005), 153-160.S. Lin, Point-countable covers and sequence-covering mappings, Science Press, Beijing, 2015 (in Chinese).S. Lin and Z.Q. Yun, Generalized metric spaces and mapping, Atlantis Studies in Mathematics 6, Atlantis Press, Paris, 2016. https://doi.org/10.2991/978-94-6239-216-8S. K. Pal, N. Adhikary and U. Samanta, On ideal sequence covering maps, Appl. Gen. Topol. 20, no. 2 (2019), 363-377. https://doi.org/10.4995/agt.2019.11238H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73-74. https://doi.org/10.4064/cm-2-2-98-108Z. Tang and F. Lin, Statistical versions of sequential and Fréchet-Urysohn spaces, Adv. Math. (China) 44 (2015), 945-954.X. G. Zhou and M. Zhang, More about the kernel convergence and the ideal convergence, Acta Math. Sinica, English Series 29 (2013), 2367-2372.X. G. Zhou and L. liu, On I-covering mappings and 1-I-covering mappings, J. Math. Res. Appl. (China) 40, no. 1 (2020) 47-56.X. G. Zhou, L. Liu and S. Lin, On topological spaces defined by I-convergence, Bull. Iran. Math. Soc. 46 (2020), 675-692. https://doi.org/10.1007/s41980-019-00284-

On real-valued measures of statistical type and their applications to statistical convergence

NSFC [10771175]A real-valued finitely additive measure mu on N is said to be a measure of statistical type provided mu(k) = 0 for all singletons {k}. Applying the classical representation theorem of finitely additive measures with totally bounded variation, we first present a short proof of the representation theorem of statistical measures. As its application, we show that every kind of statistical convergence is just a type of measure convergence with respect to a specific class of statistical measures. (C) 2009 Elsevier Ltd. All rights reserved