Abstract
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 is devoted to the discussion of fuzzy rough sets and their topological structures. Fuzzy rough approximations are further investigated. Fuzzy relations are researched by means of topology or lower and upper sets. Topological structures of fuzzy approximation spaces are given by means of pseudoconstant fuzzy relations. Fuzzy topology satisfying (CC) axiom is investigated. The fact that there exists a one-to-one correspondence between the set of all preorder fuzzy relations and the set of all fuzzy topologies satisfying (CC) axiom is proved, the concept of fuzzy approximating spaces is introduced, and decision conditions that a fuzzy topological space is a fuzzy approximating space are obtained, which illustrates that we can research fuzzy relations or fuzzy approximation spaces by means of topology and vice versa. Moreover, fuzzy pseudoclosure operators are examined.
Highlights
Rough set theory, proposed by Pawlak [1], is a new mathematical tool for data reasoning
We proved that there exists a one-to-one correspondence between the set of all preorder fuzzy relations and the set of all fuzzy topologies which satisfy (CC) axiom
We presented the concept of pseudoconstant fuzzy relations and gave topological structures of fuzzy approximation spaces by using them
Summary
Rough set theory, proposed by Pawlak [1], is a new mathematical tool for data reasoning. The equivalence relation is, too restrictive for many practical applications To address this issue, many interesting and meaningful extensions of Pawlak rough sets have been presented. The other one is the axiomatic approach in which the lower and upper approximation operators satisfy a set of axioms that are the same as those constructed ones. It often happens that some attribute values for an object of an information system are missing For such incomplete information systems, Kryszkiewicz [32, 33] and Salama [34] studied the rule generation and information recovery by rough set approach and topological method, respectively.
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