Skip to main content
SpringerLink
Log in

Rough sets and topological spaces based on similarity

  • Original Article
  • Published:
International Journal of Machine Learning and Cybernetics Aims and scope Submit manuscript

Abstract

Ordinary topology now has been used in many sub-fields of artificial intelligence, such as knowledge representation, spatial reasoning etc. In this paper, we discuss the relationships between the theory of rough sets and topological spaces. We obtain the concepts of rough sets from topology, also we obtain the concepts of topological space from rough sets. Furthermore, if the function between two topological spaces is bijective, open and continuous, then the image and the inverse image of a rough set are also rough.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abo-Tabl EA (2008) Some topological concepts and their applications from the relations point of view, PhD. Thesis-Assiut University, Egypt

  2. Abo-Tabl EA (2011) A comparison of two kinds of definitions of rough approximations based on a similarity relation. Inf Sci 181:2587–2596

    Article  MathSciNet  MATH  Google Scholar 

  3. Abu-Donia HM (2012) Multi knowledge based rough approximations and applications. Knowl Based Syst 26:20–29

    Article  Google Scholar 

  4. Allam AA, Bakeir MY, Abo-Tabl EA (2011) Approximation space on novel granulations. J Egypt Math Soc (accepted)

  5. Bonikowski Z, Bryniarski E, Wybraniec-Skardowska U (1998) Extensions and intentions in the rough set theory. Inf Sci 107:149–167

    Article  MathSciNet  MATH  Google Scholar 

  6. Chakrabarty K, Biswas R, Nanda S (2000) Fuzziness in rough sets. Fuzzy Sets Syst 110:247–251

    Article  MathSciNet  MATH  Google Scholar 

  7. Fan T, Pei D (2007) Topological description of rough set models. In: Proceedings of the IEEE international conference on fuzzy sets and knowledge discovery 2007, Haikou, China, August 24–27, vol 3, pp 229–232

  8. Hoehle U (1988) Quotients with respect to similarity relations. Fuzzy Sets Syst 27:31–44

    Article  MATH  Google Scholar 

  9. Jarvinen J, Kortelainen J (2007) A unifying study between model-like operators, topologies, and fuzzy sets. Fuzzy Sets Syst 158:1217–1225

    Article  MathSciNet  Google Scholar 

  10. Kondo M (2006) On the structure of generalized rough sets. Inf Sci 176:589–600

    Article  MathSciNet  MATH  Google Scholar 

  11. Kortelainen J (1994) On relationship between modified sets. topological spaces and rough sets. Fuzzy Sets Syst 61:91–95

    Article  MathSciNet  MATH  Google Scholar 

  12. Lashin E, Kozae A, Khadra AA, Medhat T (2005) Rough set theory for topological spaces. Int J Approx Reason 40(1–2):35–43

    Article  MATH  Google Scholar 

  13. Li X, Liu S (2012) Matroidal approaches to rough sets via closure operators. Int J Approx Reason 53(4):513–527

    Article  MATH  Google Scholar 

  14. Li Z, Xie T, Li Q (2012) Topological structure of generalized rough sets. Comput Math Appl 63:1066–1071

    Article  MathSciNet  MATH  Google Scholar 

  15. Li T-J, Yeung Y, Zhang W-X (2008) Generalized fuzzy rough approximation operators based on fuzzy covering. Int J Approx Reason 48:836–856

    Article  MATH  Google Scholar 

  16. Lin TY (1992) Topological and fuzzy rough sets. In: Slowinski R (ed) Intelligent decision support, Kluwer, Dordrecht, pp 287–304

  17. Lin TY (1998) Granular computing on binary relations (I). In: Polkowski L, Skowron A (eds) Rough sets in knowledge discovery, vol. 1, Physica-Verlag, Heidelberg, pp 107–121

  18. Liu GL, Zhu W (2008) The algebraic structures of generalized rough set theory. Inf Sci 178(21):4105–4113

    Article  MATH  Google Scholar 

  19. Nguyen HT (2000) Some mathematical structures for computational information. Inf Sci 128:67–89

    Article  MATH  Google Scholar 

  20. Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356

    Article  MathSciNet  MATH  Google Scholar 

  21. Pawlak Z (1991) Rough sets: theoretical aspects of reasoning about data. Kluwer Academic Publishers, Boston

  22. Pei D (2007) On definable concepts of rough set models. Inf Sci 177:4230–4239

    Article  MATH  Google Scholar 

  23. Pei Z, Pei D, Zheng L (2011) Topology vs generalized rough sets. Int J Approx Reason 52:231–239

    Article  MathSciNet  MATH  Google Scholar 

  24. Polkowski L (2001) On fractals defined in information systems via rough set theory. In: Proceedings of the RSTGC-2001. Bull Int Rough Set Society, vol 5(1/2), pp 163–166

  25. Polkowski L (2002) Rough sets: mathematical foundations. Physica-Verlag, Heidelberg

    Book  Google Scholar 

  26. Qin K, Pei Z (2005) On the topological properties of fuzzy rough sets. Fuzzy Sets Syst 151:601–613

    Article  MathSciNet  MATH  Google Scholar 

  27. Skowron A (1988) On topology in information system. Bull Pol Acad Sci Math 36:477–480

    MathSciNet  MATH  Google Scholar 

  28. Skowron A, Stepaniuk J (1996) Tolerance approximation spaces. Fundamenta Informaticae 27:245–253

    MathSciNet  MATH  Google Scholar 

  29. Skowron A, Swiniarski R, Synak P (2005) Approximation spaces and information granulation. Trans Rough Sets III Lect Notes Comput Sci 3400:175–189

    Article  Google Scholar 

  30. Slowinski R, Vanderpooten D (2000) A generalized definition of rough approximations based on similarity. IEEE Trans Knowl Data Eng 12(2):331–336

    Article  Google Scholar 

  31. Sierpenski W, Krieger C (1956) General topology. University of Toronto, Toronto

    Google Scholar 

  32. Srivastava AK, Tiwari SP (2003) On relationships among fuzzy approximation operators, fuzzy topology, and fuzzy automata. Fuzzy Sets Syst 138:197–204

    Article  MathSciNet  MATH  Google Scholar 

  33. Thuan ND (2009) Covering rough sets from a topological point of view. Int J Comput Theory Eng 1(5):1793–8201

    Google Scholar 

  34. Tversky A (1977) Features of similarity. Psychol Rev 84(4):327–352

    Article  Google Scholar 

  35. Wiweger A (1988) On topological rough sets. Bull Pol Acad Sci Math 37:51–62

    Google Scholar 

  36. Yang X, Song X, Chen Z, Yang J (2011) On multigranulation rough sets in incomplete information systems. Int J Mach Learn Cybern. doi:10.1007/s13042-011-0054-8

  37. Yang L, Xu L (2011) Topological properties of generalized approximation spaces. Inf Sci 181:3570–3580

    Article  MATH  Google Scholar 

  38. Yao YY (1998) Relational interpretations of neighborhood operators and rough set approximation operators. Inf Sci 111(1–4):239–259

    Article  MATH  Google Scholar 

  39. Zhu W (2007) Topological approaches to covering rough sets. Inf Sci 177(6):1499–1508

    Article  MATH  Google Scholar 

  40. Zhu W (2007) Generalized rough sets based on relations. Inf Sci 177(22):4997–5011

    Article  MATH  Google Scholar 

  41. Zhu W (2009) Relationship between generalized rough sets based on binary relation and covering. Inf Sci 179(3):210–225

    Article  MATH  Google Scholar 

  42. Zhu W, Wang S (2011) Matroidal approaches to generalized rough sets based on relations. Int J Mach Learn Cybern 2(4):273–279

    Article  Google Scholar 

  43. Zhu W, Wang F-Y (2003) Reduction and axiomization of covering generalized rough sets. Inf Sci 152(1):217–230

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The author would like to thank the anonymous referees for their valuable suggestions in improving this paper.

Author information

Authors and Affiliations

  1. Mathematics Department, Faculty of Science, Assiut University, Assiut, 71516, Egypt

    E. A. Abo-Tabl

  2. Mathematics Department, Faculty of Science and Arts, Al-mithnab, Qassim University, P.O. Box 931, Buraydah, 51931, Al-mithnab, Kingdom of Saudi Arabia

    E. A. Abo-Tabl

Authors
  1. E. A. Abo-Tabl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. A. Abo-Tabl.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abo-Tabl, E.A. Rough sets and topological spaces based on similarity. Int. J. Mach. Learn. & Cyber. 4, 451–458 (2013). https://doi.org/10.1007/s13042-012-0107-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13042-012-0107-7

Keywords

Search

Navigation