Topological properties of generalized approximation spaces. (English) Zbl 1242.68342

Summary: Rough set theory is a powerful mathematical tool for dealing with inexact, uncertain or vague information. The core concepts of rough set theory are information systems and approximation operators of approximation spaces. Approximation operators draw close links between rough set theory and topology. This paper concerns generalized approximation spaces via topological methods and studies topological properties of rough sets. Classical separation axioms, compactness and connectedness for topological spaces are extended to generalized approximation spaces. Relationships among separation axioms for generalized approximation spaces and relationships between topological spaces and their induced generalized approximation spaces are investigated. An example is given to illustrate a new approach to recover missing values for incomplete information systems by regularity of generalized approximation spaces.


68T37 Reasoning under uncertainty in the context of artificial intelligence
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D30 Compactness
54D05 Connected and locally connected spaces (general aspects)
Full Text: DOI


[1] Aiello, M.; Van Benthem, J.; Bezhanishvili, G., Reasoning about space: the modal way, Journal of Logic and Computation, 13, 889-920 (2003) · Zbl 1054.03015
[2] Chen, D. G.; Zhang, W. X., Rough sets and topological spaces, Journal of Xi’an Jiaotong University, 35, 12, 1313-1315 (2001), (in Chinese) · Zbl 1003.54005
[3] Engelking, R., General Topology (1977), Polish Scientific Publishers: Polish Scientific Publishers Warszawa
[4] Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S., Continuous Lattices and Domains (2003), Cambridge University Press
[5] Johnstone, P. T., Stone Spaces (1982), Cambridge University Press
[6] Kondo, M., On the structure of generalized rough sets, Information Sciences, 176, 589-600 (2006) · Zbl 1096.03065
[7] Kortelainen, J., On the relationship between modified sets, topological spaces and rough sets, Fuzzy Sets and Systems, 61, 91-95 (1994) · Zbl 0828.04002
[8] Kryszkiewicz, M., Rough set approach to incomplete information systems, Information Sciences, 112, 39-49 (1998) · Zbl 0951.68548
[9] Kryszkiewicz, M., Rules in incomplete information systems, Information Sciences, 113, 271-292 (1999) · Zbl 0948.68214
[10] Lashin, E. F.; Kozae, A. M.; Abo Khadra, A. A.; Medhat, T., Rough set theory for topological spaces, International Journal of Approximate Reasoning, 40, 35-43 (2005) · Zbl 1099.68113
[11] Lin, T. Y.; Liu, Q., Rough approximate operators: axiomatic rough set theory, (Ziarko, W., Rough Sets, Fuzzy Sets and Knowledge Discovery (1994), Springer: Springer Berlin), 256-260 · Zbl 0818.03028
[12] Liu, G. L., Generalized rough sets over fuzzy lattices, Information Sciences, 178, 6, 1651-1662 (2008) · Zbl 1136.03328
[13] Liu, G. L.; Zhu, W., The algebraic structures of generalized rough set theory, Information Sciences, 178, 4105-4113 (2008) · Zbl 1162.68667
[14] Pawlak, Z., Rough sets, International Journal of Computer and Information Sciences, 11, 341-356 (1982) · Zbl 0501.68053
[15] Pawlak, Z., Rough Sets: Theoretical Aspects of Reasoning About Data (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0758.68054
[16] Pawlak, Z.; Skowron, A., Rudiments of rough sets, Information Sciences, 177, 3-27 (2007) · Zbl 1142.68549
[17] Pawlak, Z.; Skowron, A., Rough sets: some extensions, Information Sciences, 177, 28-40 (2007) · Zbl 1142.68550
[18] Pei, D. W., On definable concepts of rough set models, Information Sciences, 177, 4230-4239 (2007) · Zbl 1126.68076
[19] Qin, K. Y.; Pei, Z., On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems, 151, 3, 601-613 (2005) · Zbl 1070.54006
[20] Qin, K. Y.; Yang, J. L.; Pei, Z., Generalized rough sets based on reflexive and transitive relations, Information Sciences, 178, 4138-4141 (2008) · Zbl 1153.03316
[21] Salama, A. S., Topological solution of missing attribute values problem in incomplete information tables, Information Sciences, 180, 631-639 (2010)
[22] Wu, Q. E.; Wang, T.; Huang, Y. X.; Li, J. S., Topology theory on rough sets, IEEE Transactions on Systems, Man, and Cybernetics - Part B: Cybernetics, 38, 1, 68-77 (2008)
[23] Wu, W. Z., A study on relationship between fuzzy rough approximation operators and fuzzy topological spaces, (Wang, L.; Jin, Y., FSKD 2005. FSKD 2005, LNAI, vol. 3613 (2005), Springer-Verlag: Springer-Verlag Berlin, Heidelberg), 167-174
[24] Yao, Y. Y., Constructive and algebraic methods of the theory of rough sets, Information Sciences, 109, 21-47 (1998) · Zbl 0934.03071
[25] Yao, Y. Y., Neighborhood systems and approximate retrieval, Information Sciences, 176, 3431-3452 (2006) · Zbl 1119.68074
[26] Zhang, H. P.; Ouyang, Y.; Wang, Z. D., Note on Generalized rough sets based on reflexive and transitive relations, Information Sciences, 179, 471-473 (2009) · Zbl 1159.03328
[27] Zhang, W. X.; Wu, W. Z.; Liang, J. Y.; Li, D. Y., Rough Sets Theory and Methods (2001), Science Press: Science Press Beijing, (in Chinese)
[28] Zhu, W., Topological approaches to covering rough sets, Information Sciences, 177, 1499-1508 (2007) · Zbl 1109.68121
[29] Zhu, W., Generalized rough sets based on relations, Information Sciences, 177, 4997-5011 (2007) · Zbl 1129.68088
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