http://hdl.handle.net/11320/3664 Mon, 01 Jun 2026 17:21:05 GMT 2026-06-01T17:21:05Z http://hdl.handle.net/11320/3706 Tytuł: Topological Interpretation of Rough Sets Autorzy: Grabowski, Adam Abstrakt: 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]. Wed, 01 Jan 2014 00:00:00 GMT http://hdl.handle.net/11320/3706 2014-01-01T00:00:00Z http://hdl.handle.net/11320/3702 Tytuł: Double Series and Sums Autorzy: Endou, Noboru Abstrakt: In this paper the author constructs several properties for double series and its convergence. The notions of convergence of double sequence have already been introduced in our previous paper [18]. In section 1 we introduce double series and their convergence. Then we show the relationship between Pringsheim-type convergence and iterated convergence. In section 2 we study double series having non-negative terms. As a result, we have equality of three type sums of non-negative double sequence. In section 3 we show that if a non-negative sequence is summable, then the sequence of rearrangement of terms is summable and it has the same sums. In the last section two basic relations between double sequences and matrices are introduced. Wed, 01 Jan 2014 00:00:00 GMT http://hdl.handle.net/11320/3702 2014-01-01T00:00:00Z http://hdl.handle.net/11320/3703 Tytuł: Dual Spaces and Hahn-Banach Theorem Autorzy: Narita, Keiko; Endou, Noboru; Shidama, Yasunari Abstrakt: In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of real linear spaces. In addition, we proved properties of the norm about elements of dual spaces. For the proof we referred to descriptions in the article [21]. Finally, applying theorems of the second section, we proved the Hahn-Banach extension theorem in real normed spaces. We have used extensively used [17]. Wed, 01 Jan 2014 00:00:00 GMT http://hdl.handle.net/11320/3703 2014-01-01T00:00:00Z http://hdl.handle.net/11320/3704 Tytuł: Semiring of Sets Autorzy: Coghetto, Roland Abstrakt: Schmets [22] has developed a measure theory from a generalized notion of a semiring of sets. Goguadze [15] has introduced another generalized notion of semiring of sets and proved that all known properties that semiring have according to the old definitions are preserved. We show that this two notions are almost equivalent. We note that Patriota [20] has defined this quasi-semiring. We propose the formalization of some properties developed by the authors. Wed, 01 Jan 2014 00:00:00 GMT http://hdl.handle.net/11320/3704 2014-01-01T00:00:00Z