Characterizations of ideal cluster points

Paolo Leonetti; Fabio Maccheroni

doi:10.1515/anly-2019-0001

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Characterizations of ideal cluster points

Paolo Leonetti und Fabio Maccheroni

Veröffentlicht/Copyright: 26. Februar 2019

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Abstract

Given an ideal ℐ on ω, we prove that a sequence in a topological space X is ℐ-convergent if and only if there exists a “big” ℐ-convergent subsequence. Then we study several properties and show two characterizations of the set of ℐ-cluster points as classical cluster points of a filter on X and as the smallest closed set containing “almost all” the sequence. As a consequence, we obtain that the underlying topology τ coincides with the topology generated by the pair (τ,ℐ).

Keywords: Ideal cluster point; ideal convergence; G-ideal; filter base; statistical convergence

MSC 2010: : 40A35; 54A20; : 11B05

Acknowledgements

The authors are grateful to Szymon Głab (Łódź University of Technology, PL) and Ondřej Kalenda (Charles University, Prague) for several useful comments.

References

[1] H. Albayrak and S. Pehlivan, Statistical convergence and statistical continuity on locally solid Riesz spaces, Topology Appl. 159 (2012), no. 7, 1887–1893. 10.1016/j.topol.2011.04.026Suche in Google Scholar

[2] M. Balcerzak, K. Dems and A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007), no. 1, 715–729. 10.1016/j.jmaa.2006.05.040Suche in Google Scholar

[3] M. Balcerzak, S. Gła̧b and A. Wachowicz, Qualitative properties of ideal convergent subsequences and rearrangements, Acta Math. Hungar. 150 (2016), no. 2, 312–323. 10.1007/s10474-016-0644-8Suche in Google Scholar

[4] M. Balcerzak and P. Leonetti, On the relationship between ideal cluster points and ideal limit points, Topology Appl. 252 (2019), 178–190. 10.1016/j.topol.2018.11.022Suche in Google Scholar

[5] P. Barbarski, R. Filipów, N. Mrożek and P. Szuca, Uniform density u and ℐu-convergence on a big set, Math. Commun. 16 (2011), no. 1, 125–130. Suche in Google Scholar

[6] N. Bourbaki, General Topology. Chapters 1–4, Elem. Math. (Berlin), Springer, Berlin, 1998. Suche in Google Scholar

[7] J. Connor, The statistical and strong p-Cesàro convergence of sequences, Analysis 8 (1988), no. 1–2, 47–63. 10.1524/anly.1988.8.12.47Suche in Google Scholar

[8] J. Connor and J. Kline, On statistical limit points and the consistency of statistical convergence, J. Math. Anal. Appl. 197 (1996), no. 2, 392–399. 10.1006/jmaa.1996.0027Suche in Google Scholar

[9] P. Das, Some further results on ideal convergence in topological spaces, Topology Appl. 159 (2012), no. 10–11, 2621–2626. 10.1016/j.topol.2012.04.007Suche in Google Scholar

[10] G. Di Maio and L. D. R. Kočinac, Statistical convergence in topology, Topology Appl. 156 (2008), no. 1, 28–45. 10.1016/j.topol.2008.01.015Suche in Google Scholar

[11] I. Farah, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem. Amer. Math. Soc. 148 (2000), no. 702, 1–177. 10.1090/memo/0702Suche in Google Scholar

[12] R. Filipów, N. Mrożek, I. Recław and P. Szuca, Ideal convergence of bounded sequences, J. Symb. Log. 72 (2007), no. 2, 501–512. 10.2178/jsl/1185803621Suche in Google Scholar

[13] A. R. Freedman and J. J. Sember, Densities and summability, Pacific J. Math. 95 (1981), no. 2, 293–305. 10.2140/pjm.1981.95.293Suche in Google Scholar

[14] J. A. Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301–313. 10.1524/anly.1985.5.4.301Suche in Google Scholar

[15] J. A. Fridy, Statistical limit points, Proc. Amer. Math. Soc. 118 (1993), no. 4, 1187–1192. 10.1090/S0002-9939-1993-1181163-6Suche in Google Scholar

[16] J. A. Fridy and J. Li, Matrix transformations of statistical cores of complex sequences, Analysis (Munich) 20 (2000), no. 1, 15–34. Suche in Google Scholar

[17] J. A. Fridy and C. Orhan, Statistical limit superior and limit inferior, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3625–3631. 10.1090/S0002-9939-97-04000-8Suche in Google Scholar

[18] A. Güncan, M. A. Mamedov and S. Pehlivan, Statistical cluster points of sequences in finite dimensional spaces, Czechoslovak Math. J. 54(129) (2004), no. 1, 95–102. 10.1023/B:CMAJ.0000027250.19041.72Suche in Google Scholar

[19] P. Kostyrko, M. Mačaj, T. Šalát and O. Strauch, On statistical limit points, Proc. Amer. Math. Soc. 129 (2001), no. 9, 2647–2654. 10.1090/S0002-9939-00-05891-3Suche in Google Scholar

[20] P. Kostyrko, T. Šalát and W. A. A. Wilczyński, ℐ-convergence, Real Anal. Exchange 26 (2000/01), no. 2, 669–685. 10.2307/44154069Suche in Google Scholar

[21] A. Kwela and J. Tryba, Homogeneous ideals on countable sets, Acta Math. Hungar. 151 (2017), no. 1, 139–161. 10.1007/s10474-016-0669-zSuche in Google Scholar

[22] P. Leonetti, Continuous projections onto ideal convergent sequences, Results Math. 73 (2018), no. 3, Article ID 114. 10.1007/s00025-018-0876-8Suche in Google Scholar

[23] P. Leonetti, Thinnable ideals and invariance of cluster points, Rocky Mountain J. Math. 48 (2018), no. 6, 1951–1961. 10.1216/RMJ-2018-48-6-1951Suche in Google Scholar

[24] P. Leonetti, Invariance of ideal limit points, Topology Appl. 252 (2019), 169–177. 10.1016/j.topol.2018.11.016Suche in Google Scholar

[25] P. Leonetti, H. I. Miller and L. Miller van Wieren, Duality between measure and category of almost all subsequences of a given sequence, Period. Math. Hungar. (2018), 10.1007/s10998-018-0255-y. 10.1007/s10998-018-0255-ySuche in Google Scholar

[26] N. Levinson, Gap and Density Theorems, Amer. Math. Soc. Colloq. Publ. 26, American Mathematical Society, New York, 1940. 10.1090/coll/026Suche in Google Scholar

[27] I. J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 1, 141–145. 10.1017/S0305004100065312Suche in Google Scholar

[28] M. A. Mamedov and S. Pehlivan, Statistical cluster points and turnpike theorem in nonconvex problems, J. Math. Anal. Appl. 256 (2001), no. 2, 686–693. 10.1006/jmaa.2000.7061Suche in Google Scholar

[29] M. Marinacci, An axiomatic approach to complete patience and time invariance, J. Econom. Theory 83 (1998), no. 1, 105–144. 10.1006/jeth.1997.2451Suche in Google Scholar

[30] H. I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1811–1819. 10.1090/S0002-9947-1995-1260176-6Suche in Google Scholar

[31] F. Móricz, Statistical convergence of multiple sequences, Arch. Math. (Basel) 81 (2003), no. 1, 82–89. 10.1007/s00013-003-0506-9Suche in Google Scholar

[32] M. Mursaleen and O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003), no. 1, 223–231. 10.1007/978-81-322-1611-7_7Suche in Google Scholar

[33] A. Nabiev, S. Pehlivan and M. Gürdal, On ℐ-Cauchy sequences, Taiwanese J. Math. 11 (2007), no. 2, 569–576. Suche in Google Scholar

[34] F. Nuray and W. H. Ruckle, Generalized statistical convergence and convergence free spaces, J. Math. Anal. Appl. 245 (2000), no. 2, 513–527. 10.1006/jmaa.2000.6778Suche in Google Scholar

[35] G. Pólya, Untersuchungen über Lücken und Singularitäten von Potenzreihen, Math. Z. 29 (1929), no. 1, 549–640. 10.1007/BF01180553Suche in Google Scholar

Received: 2018-12-30

Accepted: 2019-02-04

Published Online: 2019-02-26

Published in Print: 2019-03-01

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Artikel in diesem Heft

https://doi.org/10.1515/anly-2019-0001

Schlagwörter für diesen Artikel

Ideal cluster point; ideal convergence; G-ideal; filter base; statistical convergence