Topological interpretation of rough sets. (English) Zbl 1298.54006

Summary: Rough sets, developed by Pawlak, are an important model of incomplete or partially known information. In this article, which is essentially a continuation of [A. Grabowski, “Basic properties of rough sets and rough membership function”, Formaliz. Math. 12, No. 1, 21–28 (2004)], we characterize rough sets in terms of topological closure and interior, as the approximations have the properties of the Kuratowski operators. We decided to merge topological spaces with tolerance approximation spaces. As a testbed for our developed approach, we restated the results of Y. Isomichi [Pac. J. Math. 38, 657–668 (1971; Zbl 0227.54001)] (formalized in Mizar in [M. Jastrzębska and A. Grabowski, “The properties of supercondensed sets, subcondensed sets and condensed sets”, Formaliz. Math. 13, No. 2, 353–359 (2005)]) and about fourteen sets of C. Kuratowski [Fundam. Math. 3, 182–199 (1922; JFM 48.0210.04)] (encoded with the help of Mizar adjectives and clusters’ registrations in [L. K. Bagińska and A. Grabowski, “On the Kuratowski closure-complement problem”, Formaliz. Math. 11, No. 3, 323–329 (2003)]) in terms of rough approximations. The upper bounds which were 14 and 7 in the original paper of Kuratowski, in our case are six and three, respectively. {
} It turns out that within the classification given by Isomichi, 1st class subsets are precisely crisp sets, 2nd class subsets are proper rough sets, and there are no 3rd class subsets in topological spaces generated by approximations. Also the important results about these spaces is that they are extremally disconnected [Z. Karno, “The lattice of domains of an extremally disconnected space”, Formaliz. Math. 3, No. 2, 143–149 (1992)], hence lattices of their domains are Boolean. {
} Furthermore, we develop the theory of abstract spaces equipped with maps possessing characteristic properties of rough approximations which enables us to freely use the notions from the theory of rough sets and topological spaces formalized in the Mizar Mathematical Library [A. Grabowski, Fundam. Inform. 128, No. 1–2, 65–79 (2013; Zbl 1285.68180)].


54A40 Fuzzy topology
54H10 Topological representations of algebraic systems
68T37 Reasoning under uncertainty in the context of artificial intelligence
03B35 Mechanization of proofs and logical operations


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