1. bookVolume 22 (2014): Issue 1 (March 2014)
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Journal
eISSN
1898-9934
First Published
09 Jun 2008
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4 times per year
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English
access type Open Access

Topological Interpretation of Rough Sets

Published Online: 30 Mar 2014
Page range: 89 - 97
Journal Details
License
Format
Journal
eISSN
1898-9934
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English
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 [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].

Keywords

[1] Lilla Krystyna Baginska and Adam Grabowski. On the Kuratowski closure-complement problem. Formalized Mathematics, 11(3):323-329, 2003.Search in Google Scholar

[2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Search in Google Scholar

[3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Search in Google Scholar

[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Search in Google Scholar

[5] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Search in Google Scholar

[6] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Search in Google Scholar

[7] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Search in Google Scholar

[8] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Search in Google Scholar

[9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Search in Google Scholar

[10] Adam Grabowski. Automated discovery of properties of rough sets. Fundamenta Informaticae, 128:65-79, 2013. doi:10.3233/FI-2013-933.10.3233/FI-2013-933Search in Google Scholar

[11] Adam Grabowski. Basic properties of rough sets and rough membership function. Formalized Mathematics, 12(1):21-28, 2004.Search in Google Scholar

[12] Adam Grabowski. Relational formal characterization of rough sets. Formalized Mathematics, 21(1):55-64, 2013. doi:10.2478/forma-2013-0006.10.2478/forma-2013-0006Search in Google Scholar

[13] Yoshinori Isomichi. New concepts in the theory of topological space - supercondensed set, subcondensed set, and condensed set. Pacific Journal of Mathematics, 38(3):657-668, 1971.10.2140/pjm.1971.38.657Search in Google Scholar

[14] Magdalena Jastrz¸ebska and Adam Grabowski. The properties of supercondensed sets, subcondensed sets and condensed sets. Formalized Mathematics, 13(2):353-359, 2005.Search in Google Scholar

[15] Zbigniew Karno. The lattice of domains of an extremally disconnected space. Formalized Mathematics, 3(2):143-149, 1992.Search in Google Scholar

[16] Artur Korniłowicz. On the topological properties of meet-continuous lattices. Formalized Mathematics, 6(2):269-277, 1997.Search in Google Scholar

[17] Kazimierz Kuratowski. Sur l’opération A de l’analysis situs. Fundamenta Mathematicae, 3:182-199, 1922.10.4064/fm-3-1-182-199Search in Google Scholar

[18] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Search in Google Scholar

[19] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Search in Google Scholar

[20] Bartłomiej Skorulski. First-countable, sequential, and Frechet spaces. Formalized Mathematics, 7(1):81-86, 1998.Search in Google Scholar

[21] Bartłomiej Skorulski. The sequential closure operator in sequential and Frechet spaces. Formalized Mathematics, 8(1):47-54, 1999.Search in Google Scholar

[22] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Search in Google Scholar

[23] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1 (3):495-500, 1990.Search in Google Scholar

[24] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1 (1):187-190, 1990.Search in Google Scholar

[25] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics, 1(2):387-393, 1990.Search in Google Scholar

[26] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Search in Google Scholar

[27] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Search in Google Scholar

[28] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231-237, 1990. Search in Google Scholar

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