Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) https://internationalpubls.com 1149 Number Of Distinct Topologies On Finite Set With R-Open Sets Dr. Narendrakumar R. Dasre1, Dr. Pritam Gujarathi-Wani2, Dr. Manohar B. Bhagirath3* 1Associate Professor, Head, Department of Engineering Sciences, Ramrao Adik Institute of Technology, Nerul, Navi Mumbai-400706 Email-id : narendasre@rait.ac.in 2Assistant Professor, Department of Engineering Sciences, Ramrao Adik Institute of Technology, Nerul, Navi Mumbai-400706 Email-id : pritam.wani@rait.ac.in 3*Associate Professor, Head, Department of Mathematics, Annasaheb Vartak College of Arts, Science and Commerce, Vasai, Dist. Palghar - 401202. Email-id : manoharbhagirath@gmail.com Article History: Received:11-02-2025 Revised: 24-02-2025 Accepted:11-03-2025 Abstract: In this paper, we have studied the sequences of distinct topologies on n-set with r-open sets [2]. Here we have applied the concept of repeated ratios proposed by Dasre and Gujarathi [3] to these sequences. After applying the repeated ratios to these sequences, we observed that the repeated ratios follow the same pattern for all sequences. So the method of repeated ratios can be used for finding the better bounds for the number of topologies. Using the repeated ratios, one can approximate the large numbers for the sequences as in [3, 4]. We also observed that the sequence of repeated ratios oscillates around one. AMS Subject Classification: 40A15, 54A99. Key Words and Phrases: Topology, Repeated Ratio, Sequences. 1. Introduction In literature very few approaches are available for finding number of topologies on larger finite set. The traditional method for finding number of topologies on finite sets are found insufficient for large n. The approaches by V. Krishnamoorthy [5], G. H. Patil and M. S. Chaudhary [1] have tried to give bounds but still these bounds are not efficient for large n. The method introduced by Dasre and Gujarathi of repeated ratios in [3] is applicable for large n. Also by the same approach in [4] the method was applied to partially ordered set with n-labelled elements and have given the approximate bounds. In this paper, we have applied the method of repeated ratios to the series of topologies on n- sets with r-open sets. 2. Repeated Ratios Let {𝑥𝑛} be the series. We define the repeated ratios [3] as below: 𝑎𝑛 = 𝑋𝑛+1 𝑋𝑛 … . (1) 𝑏𝑛 = 𝑎𝑛+1 𝑎𝑛 = 𝑋𝑛+2𝑋𝑛 𝑋𝑛+1 2 … . (2) 𝑐𝑛 = 𝑏𝑛+1 𝑏𝑛 = 𝑋𝑛+3𝑋𝑛+1 3 𝑋𝑛+2 3 𝑋𝑛 … . (3) and so on. In this paper, we are considering the series A281773, A281774, A281775, A281776, A281777, A281778, A281779, A281780 of topologies on sets of n-elements with r-open sets by N. J. Sloane[2]. The repeated ratios an, bn, cn, . . . , jn are as defined in [3]. These repeated ratios are calculated for series of distinct topologies on n-sets with exactly r-open sets. We have tabulated the repeated ratios for each of the above series [2] as below: The series A281773: 0, 0, 1, 9, 43, 165, 571, 1869, 5923, 18405, 56491, 172029, 521203, 1573845, 4742011, 14266989, 42882883, 128812485, 386765131, 1160950749, 3484162963, 10455110325, mailto:%20narendasre@rait.ac.in mailto:pritam.wani@rait.ac.in mailto:manoharbhagirath@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) https://internationalpubls.com 1150 31370573851, 94122207309, 282387593443, 847204723365, 2541698056171, 7625261940669 is a series of Number of distinct topologies on an n-set that have exactly 4 open sets. Table 1 shows repeated ratios of the non-zero terms of the series A281773. . The series A281774: 0, 0, 0, 6, 72, 630, 4680, 31206, 193032, 1131990, 6386760, 35025606, 188061192, 993760950, 5187840840, 26831095206, 137770476552, 703455087510, 3576115150920, 18117222864006, 91536570671112, 461496288791670, 2322770028381000, 11675109032796006 is a series of Number of distinct topologies on an n-set that have exactly 6 open sets. Table 2 shows repeated ratios of the non-zero terms of the series A281774. The series A281775: 0, 0, 0, 0, 54, 780, 7830, 67620, 535374, 3992940, 28483110, 196316340, 1317106494, 8650141500, 55853351190, 355770438660, 2241509994414, 13998294536460, 86795899256070, 535048203626580, 3282628800655134, 20061393719417820, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) https://internationalpubls.com 1151 122212221633141750 is a series of Number of distinct topologies on an n-set that have exactly 7 open sets. Table 3 shows repeated ratios of the non-zero terms of the series A281775. The series A281776: 0, 0, 0, 1, 54, 955, 11760, 122941, 1175034, 10595215, 91506420, 763624081, 6194818014, 49084747075, 381338401080, 2914184784421, 21965095364994, 163656285828535, 1207613518375740, 8838842878371961, 64253768864671974, 464416229729871595, 3340518964319750400 is a series of Number of distinct topologies on an n-set that have exactly 8 open sets. Table 4 shows repeated ratios of the non-zero terms of the series A281776. The series A281777: 0, 0, 0, 0, 20, 800, 14260, 189280, 2181060, 23241120, 235737620, 2308206560, 21979728100, 204477713440, 1864504348980, 16707856095840, 147469451067140, 1284607771225760, 11063319237792340, 94343562846289120, 797685042851814180, 6694943490279586080 is a series of Number of distinct topologies on an n-set that have exactly 9 open sets. Table 6 shows repeated ratios of the non-zero terms of the series A281777. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) https://internationalpubls.com 1152 The series A2817778 0, 0, 0, 0, 24, 900, 18030, 276570, 3680964, 45065160, 523292010, 5859909990, 63862084704, 680829769620, 7122705252390, 73284607133010, 742843170653244, 7429450873589280, 73416173732059170, 717721593866613630, 6949589106333898584, 66721599431782204140 is a series of Number of distinct topologies on an n-set that have exactly 10 open sets. Table 7 shows repeated ratios of the non-zero terms of the series A281778. The series A2817779: 0, 0, 0, 0, 0, 500, 16980, 342160, 5486040, 77926380, 1031160060, 13047426920, 160124426880, 1921105846660, 22632779709540, 262513678889280, 3002768326532520, 33914184260797340, 378596540805849420, 4181330954328313240, 45727913513193402960, 495618273676457274420 is a series of Number of distinct topologies on an n-set that have exactly 11 open sets. Table 8 shows repeated ratios of the non-zero terms of the series A281779. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) https://internationalpubls.com 1153 The series A2817780: 0, 0, 0, 0, 12, 660, 20400, 445620, 7977732, 126860580, 1873839000, 26381789940, 359484471852, 4784481401700, 62538498859200, 805447464281460, 10241415118476372, 128722997969290020, 1600670708273985000, 19705915838479512180, 240330009637668935292 is a series of Number of distinct topologies on an n-set that have exactly 12 open sets. Table 8 shows repeated ratios of the non-zero terms of the series A281780. From the above tables, one can observe that repeated ratios for all above series follow the same pattern bn > 1 , cn < 1, dn > 1, en < 1 and so on. Also one can observe that the sequence of repeated ratios {bn, cn, dn, ....} oscillates around 1 for each series. After approximating these repeated ratios to 1, we can find better upper and lower bounds for each series as in [3, 4]. Conclusion In this paper, we have calculated the repeated ratios for the series of topologies defined on n-sets with exactly r-open sets. We observed that the sequence of repeated ratios {an, bn, cn, dn, ....} oscillates around 1. Hence we propose the conjecture that the sequence of repeated ratios converges to 1 for the series of topologies defined on n-sets with exactly r-open sets. Acknowledgments We are thankful to Dr. Mukesh D. Patil, The Principal, for his continuous guidance and support throughout the research work. We would like to extend my heartfelt gratitude to Dr. Rajendra P. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) https://internationalpubls.com 1154 Deore, our respected mentor, for his invaluable guidance and unwavering support throughout this research work. References [1] G. H. Patil and M. S. Chaudhary, A recursive determination of topologies on finite sets, Indian Journal of Pure and Applied Mathematics, 26(2),(1995), 143–148. [2] N. J. A. Sloane, Online Encyclopedia of Integer Sequences, at http://oeis.org [3] N. R. Dasre and Pritam Gujarathi, Topologies on finite sets, International Journal of Pure and Applied Mathematics, 118(1), (2018), 39-48. [4] N. R. Dasre and Pritam Gujarathi, , presented at 4th International Conference on Computing in Engineering and Technology( ICCET), 9-11 Jan, 2019, Aurangabad, India. [5] V. Kishnamurthy, On the number of topologies on a finite set, The American Mathematical Monthly, 73(2), (1966), 154–157.