scholarly journals Categories of L-Fuzzy Čech Closure Spaces and L-Fuzzy Co-Topological Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1274
Author(s):  
Irina Perfilieva ◽  
Ahmed A. Ramadan ◽  
Enas H. Elkordy
Keyword(s):  
Computer Science ◽  
Fuzzy Systems ◽  
Topological Spaces ◽  
Fuzzy Relations ◽  
Approximation Spaces ◽  
Galois Correspondence ◽  
Closure Spaces

Recently, fuzzy systems have become one of the hottest topics due to their applications in the area of computer science. Therefore, in this article, we are making efforts to add new useful relationships between the selected L-fuzzy (fuzzifying) systems. In particular, we establish relationships between L-fuzzy (fuzzifying) Čech closure spaces, L-fuzzy (fuzzifying) co-topological spaces and L-fuzzy (fuzzifying) approximation spaces based on reflexive L-fuzzy relations. We also show that there is a Galois correspondence between the categories of these spaces.

M-Fuzzifying Approximation Spaces, M-Fuzzifying Pretopological Spaces and M-Fuzzifying Closure Spaces

2020 ◽  
Vol 16 (03) ◽  
pp. 609-626
Author(s):  
Anand P. Singh ◽  
I. Perfilieva
Keyword(s):  
Significant Role ◽  
Category Theory ◽  
Galois Connection ◽  
Fuzzy Relations ◽  
Approximation Spaces ◽  
Closure Spaces

In category theory, Galois connection plays a significant role in developing the connections among different structures. The objective of this work is to investigate the essential connections among several categories with a weaker structure than that of [Formula: see text]-fuzzifying topology, viz. category of [Formula: see text]-fuzzifying approximation spaces based on reflexive [Formula: see text]-fuzzy relations, category of [Formula: see text]-fuzzifying pretopological spaces and the category of [Formula: see text]-fuzzifying interior (closure) spaces. The interrelations among these structures are shown via the functorial diagram.

scholarly journals Near Approximations in -Closure Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-23
Author(s):  
M. E. Abd El-Monsef ◽  
M. Shokry ◽  
Y. Y. Yousif
Keyword(s):  
Mathematical Models ◽  
Real Life ◽  
Approximation Spaces ◽  
Closure Approximation ◽  
Closure Spaces

Most real-life situations need some sort of approximation to fit mathematical models. The beauty of using topology in approximation is achieved via obtaining approximation for qualitative subgraphs without coding or using assumption. The aim of this paper is to apply near concepts in the -closure approximation spaces. The basic notions of near approximations are introduced and sufficiently illustrated. Near approximations are considered as mathematical tools to modify the approximations of graphs. Moreover, proved results, examples, and counterexamples are provided.

scholarly journals Borel and Hausdorff hierarchies in topological spaces of Choquet games and their effectivization

2014 ◽  
Vol 25 (7) ◽  
pp. 1490-1519 ◽  
Author(s):  
VERÓNICA BECHER ◽  
SERGE GRIGORIEFF
Keyword(s):  
Topological Spaces ◽  
Descriptive Set Theory ◽  
Stationary Strategy ◽  
Approximation Spaces ◽  
Polish Spaces ◽  
Proper Subclass ◽  
Continuous Domains ◽  
Algebraic Domains ◽  
Descriptive Set ◽  
Work Done

What parts of the classical descriptive set theory done in Polish spaces still hold for more general topological spaces, possibly T0 or T1, but not T2 (i.e. not Hausdorff)? This question has been addressed by Selivanov in a series of papers centred on algebraic domains. And recently it has been considered by de Brecht for quasi-Polish spaces, a framework that contains both countably based continuous domains and Polish spaces. In this paper, we present alternative unifying topological spaces, that we call approximation spaces. They are exactly the spaces for which player Nonempty has a stationary strategy in the Choquet game. A natural proper subclass of approximation spaces coincides with the class of quasi-Polish spaces. We study the Borel and Hausdorff difference hierarchies in approximation spaces, revisiting the work done for the other topological spaces. We also consider the problem of effectivization of these results.

scholarly journals Topological Interpretation of Rough Sets

2014 ◽  
Vol 22 (1) ◽  
pp. 89-97 ◽  
Author(s):  
Adam Grabowski
Keyword(s):  
Rough Sets ◽  
Upper Bounds ◽  
Topological Spaces ◽  
Approximation Spaces ◽  
Extremally Disconnected ◽  
Topological Interpretation ◽  
Topological Closure ◽  
Mizar Mathematical Library ◽  
Abstract Spaces

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].

INTERVAL METHODS IN KNOWLEDGE REPRESENTATION

1996 ◽  
Vol 04 (04) ◽  
pp. 391-393
Author(s):  
Vladik Kreinovich
Keyword(s):  
Conference Proceedings ◽  
Knowledge Representation ◽  
Computer Science ◽  
Fuzzy Systems ◽  
International Workshop ◽  
El Paso ◽  
Annotated Bibliography ◽  
Interval Methods ◽  
Interval Computations ◽  
University Of Texas

With this issue, we start a new section: short abstract of papers and books on interval methods in knowledge representation. The experience of several conferences on interval computations and fuzzy systems, including the 1995 International Workshop on Applications of Interval Computations (February 1995, El Paso, TX, USA), has shown that there are many areas of knowledge representation where interval methods are applied, and many interesting results of these applications, areas and results that are often not widely known to the knowledge representation community. These papers are published in different journals and conference proceedings, and it is difficult to trace them all. In view of this difficulty, we decided to provide the readers of IJUFKS with short abstract of these papers (something like an ongoing annotated bibliography). We strongly believe that the information about the current applications of interval methods is of interest to this community. For the reasons expressed above, we are currently, more probably, not covering all relevant papers. To increase the coverage, we need your help. If you know of any recent papers devoted to the applications of interval methods to knowledge representation, please send reference to Vladik Kreinovich at [email protected], or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA. If you have writtend your own reviews of such papers, or if you would like to write such reviews, please contact Vladik as well. Authors, please send information and/or copies of your own applications papers (papers in French, Russian, and German are also welcome). Abstracts should ideally in LATEX or TEX, but ASCII is also acceptable This is a new section, and we want the readers' input about how to make it better. Any suggestions and recommendations will be highly welcome.

Simplicity of Categories Defined by Symmetry Axioms

1991 ◽  
Vol 34 (2) ◽  
pp. 240-248
Author(s):  
E. Lowen-Colebunders ◽  
Z. G. Szabo
Keyword(s):  
Topological Spaces ◽  
Convergence Spaces ◽  
Epireflective Subcategory ◽  
Closure Spaces ◽  
Symmetry Axiom

AbstractWe consider two generalizations R0w and R0 of the usual symmetry axiom for topological spaces to arbitrary closure spaces and convergence spaces. It is known that the two properties coincide on Top and define a non-simple subcategory. We show that R0W defines a simple subcategory of closure spaces and R0 a non-simple one. The last negative result follows from the stronger statement that every epireflective subcategory of R0 Conv containing all T1 regular topological spaces is not simple. Similar theorems are shown for the topological categories Fil and Mer.

Sign in / Sign up

Export Citation Format

Share Document