Monad Discrete Spaces and Limit Monad Points on Them

Year 2026, Volume: 22 Issue: 1, 67 - 75, 30.03.2026

Orhan Göçür

https://doi.org/10.18466/cbayarfbe.1640028

https://izlik.org/JA24CE48KT

Abstract

In this study, Monad discrete spaces were defined on amply soft topologies which is a more flexible and comprehensive structure from other known traditional soft topologies. The conditions under which a given amply soft set is amply soft open or amply soft closed set were given over them. Then, the amply soft set of limit monad points and closure points of any amply soft set in monad discrete spaces was investigated and the relevant properties were given. And then, as an extra, limit monad points in amply soft topological spaces produced by classical topologies were investigated and the relevant property was given. While constructing amply soft sets, specifically focused on amply soft topologies formed by selecting real numbers from both universal and parametric spaces unlike the traditional soft sets. Finally, a question was left that we hope will catch the reader's attention.

Keywords

a new extended soft set , limit monad point , monad discrete space , PAS topology

Ethical Statement

There are no ethical issues after the publication of this manuscript journal.

Thanks

I would like to express my gratitude to the journal's esteemed reviewers for their suggestions during the writing evaluation process, which contributed to the final version of this manuscript.

References

• [1]. Zadeh L.A. 1965. Fuzzy sets. Inform Control, 8, 338-353.
• [2]. Gorzalzany M.B. 1987. A method of inference in approximate reasoning based on interval valued fuzzy sets. Fuzzy Sets Syst. 21, 1-17.
• [3]. Atanassov K. 1986. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87-96.
• [4]. Smarandache F. 2005. Neutrosophic set, a generalisation of the generalized fuzzy sets, Inter. J.Pure Appl. Math. 24, 287-297.
• [5]. Gau W.L. and Buehrer D.J. 1993. Vague sets, IEEE Trans. System Man Cybernet, 23.2, 610- 614.
• [6]. Pawlak Z. 1982. Rough sets. Int. J. Comp. Inf. Sci. 11, 341- 356.
• [7]. Molodtsov D. 1999. Soft set theory first results. Comput. Math. Appl. 37, 19- 31.
• [8]. Maji P.K., Biswas R., Roy R. 2003. Soft set theory, Comput. Math. Appl., 45 555-562.
• [9]. Yang C.F. 2008. A note on "Soft set theory" [Computers and Mathematics with Applications 45 (2003) 555-562]. Comput. Math. Appl., 56, 1899-1900.
• [10]. Ali M.I., Feng F., Liu X., Min W.K. 2009. On some new operations in soft set theory. Comput. Math. Appl. 57, 1547-1553.
• [11]. Neog T.J., Sut D.M. 2011. A New Approach to the Theory of Soft Sets. Int. J. Comput. Appl. 32, 1-6
• [12]. Aygünoğlu A., Aygün A.H. 2012. Some notes on soft topological spaces. Neural Comp. Appl. 21, 113-119.
• [13]. Kamacı H., Atagün A.O., Aygün E. 2019. Difference Operations of Soft Matrices with Applications in Decision Making. Punjab Univ. J. Math. 51, 1-21.
• [14]. Kamacı H. 2019. Selectivity analysis of parameters in soft set and its effect on decision making. Int. J. Mach. Learn. Cyber., doi:10.1007/s13042-019-00975-w.
• [15]. Sezgın A., Atagün A.O. 2011. On operations of soft sets. Comput. Math. Appl. 61, 1457--1467.
• [16]. Sezgin A., Cagman N. 2024. A New Soft Set Operation: Complementary Soft Binary Piecewise Difference (\) Operation, Osmaniye Korkut Ata UniversityJournal of natural and applied sciences 7.1, 58-94.
• [17]. Sezgin A., Durak İ. 2025. Soft Union-difference product of groups, Universal Library of Multidisciplinary, 2,1.
• [18]. Çağman N., Karataş S., Enginoğlu S. 2011. Soft Topology. Comput. Math. Appl. 62, 351-358
• [19]. Çağman N., Enginoğlu S., Çıtak F. 2011. Fuzzy soft set theory and its applications, Iranian Journal of Fuzzy Systems, 8, 137-147.
• [20]. Jyothis T., Sunil J.J. 2016. A note on soft topology. J. New Results in Sci. 11, 24-29.
• [21]. Shabir M., Naz M. 2011. On soft topological spaces. Comput. Math. Appl. 61, 1786-1799
• [22]. Göçür, O. 2017. Soft single point space and soft metrizable. Ann. Fuzzy Math. Inform, 13, 499-507.
• [23]. Das S., Samanta S.K. 2013. Soft metric. Ann. Fuzzy Math. Inform, 6, 77-94.
• [24]. Zhu P.,Wen Q. 2013. Operations on Soft Sets Revisited. J. Appl. Math. 1, 1-7.
• [25]. Fatimah F., Rosadi D., Hakim R.B.F. 2028. N-Soft Sets and Decision Making Algorithms. Soft Comput. 22, 3829-3842.
• [26]. Smarandache F. 2018. Extension of soft set to hypersoft set, and then to plithogenic hypersoft set. Neutrosophic Sets Syst. 22, 168-170.
• [27]. Dalkılıç O., Demirtaş N. 2022. Decision analysis review on the concept of class for bipolar soft set theory, Comput. Appl. Mat. 41.5, 205.
• [28]. Musa S.Y., Asaad B.A. 2021. Bipolar hypersoft sets, Mathematics, 9, 1826.
• [29]. Musa S.Y., Mohammed R.Y., Asaad B. 2023. N-hypersoft sets: An innovative extension of hypersoft sets and their applications. Symmetry, 15.9, 1795.
• [30]. Musa S.Y. 2024. N-bipolar hypersoft sets: Enhancing decision-making algorithms. Plos one, 19.1
• [31]. Göçür O. 2021. Amply soft set and its topologies: AS and PAS topologies. AIMS Mathematics, 6.4, 3121-3141.
• [32]. Göçür O. 2020. Monad metrizable space. Mathematics, 8, 1891.
• [33]. Göçür O. 2022. on Amply soft topological spaces. World Scientific News, 171, 21-34.
• [34]. Göçür O. 2022. neighbourhoods on Amply Soft topological Spaces. World Scientific News, 172, 105-117.
• [35]. Karakaş G., Göçür O., Kopuzlu A. 2025. isolated monad points on amply soft topological spaces, Journal of the Institute of Science and Technology, 15.4, 1504-1512.
• [36]. Göçür O. 2025. limit monad points and investigating them on amply soft topological spaces, Filomat, in review.
• [37]. Kaku M. 2021. The Future of humanity, terra formating mars, intersteller Travel, and our destinity beyond earth Living in hyperspace, 3 eds., ODTÜ Press, Ankara,Turkey, 300-330.
• [38]. Altındağ Ö. 2025. Non-Parametric Inference for Multi-Sample of Geometric Processes with Application to Multi-System Repair Process Modeling. Mathematics, 13.14, 2260.