I-cluster points of filters

Year 2017, Volume: 5 Issue: 4, 195 - 202, 01.10.2017

Rohini Jamwal, Renu Renu Dalip Singh Jamwal

Abstract

In this paper, we have introduced the concept of cluster point of a filter on a topological space and studied its various properties. We have proved the necessary condition for a filter to have an cluster point. Most of the work in this paper is inspired from [2] and .

Keywords

Filters, nets, convergence, cluster point

References

• V. Bal J. erveansk P. Kostyrko, T. alt, convergence and continuity of real functions, Faculty of Natural Sciences, Constantine the Philosoper University, Nitra, Acta Mathematical 5, 43-50, 2002.
• N. Bourbaki, General Topology, Part (I) (transl.), Addison- Wesley, Reading (1966).
• K. Demirci, I-limit superior and limit inferior, Math. Commun. 6 (2001), 165-172.
• H. Fast, sur la convergence statistique, colloq. Math. 2 (1951), 241-244.
• H. Halberstem, K. F. Roth, Sequences, Springer, New York, 1993.
• D. S. Jamwal, R. Jamwal, S. Sharma, Convergence of filters, New Trends in Mathematical Sciences, 4, No. 4, (2016), 322-328.
• R. Jamwal, S. Sharma and D. S. Jamwal, Some more results on Convergence of filters, New Trends in Mathematical Sciences, 5, No. 1, (2017), 190-195.
• P. Kostyrko, T.alt, W. Wilczynski, convergence, Real Analysis, Exch. 26 (2) (2000/2001), 669-685.
• P. Kostyrko, M. Maaj, T.alt, M. Sleziak, convergence and extremal limit points, Math. Slovaca, 55 (4) (2005), 443-464.
• B. K. Lahiri, P. Das, Further results on limit superior and limit inferior, Math. Commun., 8 (2003), 151-156.
• B. K. Lahiri, P. Das, and convergence in topological spaces, Math. Bohemica, 130 (2) (2005), 153-160.
• B. K. Lahiri, P. Das, and convergence of nets, Real Analysis Exchange, 33 (2) (2007/2008), 431-442.
• M. Maaj, T.alt, Statistical convergence of subsequences of a given sequence, Math. Bohemica, 126 (2001), 191-208.
• M. Mursaleen and A. Alotaibi, On I–convergence in random 2–normed spaces, Math. Slovaca, 61(6) (2011) 933–940.
• M. Mursaleen and S. A. Mohiuddine, On ideal convergence of double sequences in probabilistic normed spaces, Math. Reports, 12(62)(4) (2010) 359-371.
• M. Mursaleen and S. A. Mohiuddine, On ideal convergence in probabilistic normed spaces, Math. Slovaca, 62(1) (2012) 49-62.
• M. Mursaleen, S. A. Mohiuddine and O. H. H. Edely, On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces, Comput. Math. Appl., 59 (2010) 603-611.
• I. Niven, H. S. Zuckerman, An introduction to the theory of numbers, 4th Ed., John Wiley, New York, 1980.
• T.alt, On statistically convergent sequences of real numbers, Mathematical Slovaca, 30 (1980), No. 2, 139-150.
• T.alt, B. C. Tripathy, M. Ziman, On convergence field, Italian J. of Pure Appl. Math. 17 (2005), 45-54.
• A.Sahiner, M. Gürdal, S. Saltan and H. Gunawan, Ideal convergence in 2-normed spaces, Taiwanese J. Math., 11(5) (2007), 1477-1484.
• I. J. Schoenberg, The integrability of certain function and related summability methods, Am. Math. Mon. 66 (1959), 361-375.
• S. Willard, General Topology, Addison-Wesley Pub. Co. 1970.