<abstract xmlns="http://www.w3.org/1999/xhtml">Rough sets, developed by Pawlak, are an important model of incomplete or partially known information. In this article, which is essentially a continuation of [11], 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 Isomichi [13] (formalized in Mizar in [14]) and about fourteen sets of Kuratowski [17] (encoded with the help of Mizar adjectives and clusters’ registrations in [1]) 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 [15], 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 [10].</abstract>

Rough sets, developed by Pawlak, are an important model of incomplete or partially known information. In this article, which is essentially a continuation of [11], 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 Isomichi [13] (formalized in Mizar in [14]) and about fourteen sets of Kuratowski [17] (encoded with the help of Mizar adjectives and clusters’ registrations in [1]) 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 [15], 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 [10].

Topological Interpretation of Rough Sets

Institute of Informatics University of Białystok Akademicka 2, 15-267 Białystok Poland

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0)

{"article-title":"Topological Interpretation of Rough Sets"}

Rough sets, developed by Pawlak, are an important model of incomplete or partially known information. In this article, which is essentially a continuation of...

Topological Interpretation of Rough Sets

Published Online: Mar 30, 2014

Page range: 89 - 97

DOI: https://doi.org/10.2478/forma-2014-0010

Keywords
rough sets, rough approximations, Kuratowski closure-complement problem, topological spaces

© by Adam Grabowski

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0)

Topological Interpretation of Rough Sets

Published Online: Mar 30, 2014

Page range: 89 - 97

DOI: https://doi.org/10.2478/forma-2014-0010

Keywordsrough sets, rough approximations, Kuratowski closure-complement problem, topological spaces

© by Adam Grabowski

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0)

Keywords
rough sets, rough approximations, Kuratowski closure-complement problem, topological spaces