Metric spaces and textures

[EN] Textures are point-set setting for fuzzy sets, and they provide a framework for the complement-free mathematical concepts. Further dimetric on textures is a gener- alization of classical metric spaces. The aim of this paper is to give some properties of dimetric texture space by using categorical approach. We prove that the category of clas- sical metric spaces is isomorphic to a full subcategory of dimetric texture spaces, and give a natural transformation from metric topologies to dimetric ditopologies. Further, it is pre- sented a relation between dimetric texture spaces and quasi-pseudo metric spaces in the sense of J. F. Kelly. Guardar / Salir Siguiente >Dost, S. (2017). Metric spaces and textures. Applied General Topology. 18(1):203-217. doi:10.4995/agt.2017.6889.SWORD203217181Brown, L. M., & Diker, M. (1998). Ditopological texture spaces and intuitionistic sets. Fuzzy Sets and Systems, 98(2), 217-224. doi:10.1016/s0165-0114(97)00358-8Brown, L. M., & Ertürk, R. (2000). Fuzzy sets as texture spaces, I. Representation theorems. Fuzzy Sets and Systems, 110(2), 227-235. doi:10.1016/s0165-0114(98)00157-2Brown, L. M., & Ertürk, R. (2000). Fuzzy sets as texture spaces, II. Subtextures and quotient textures. Fuzzy Sets and Systems, 110(2), 237-245. doi:10.1016/s0165-0114(98)00158-4Brown, L. M., Ertürk, R., & Dost, Ş. (2004). Ditopological texture spaces and fuzzy topology, I. Basic concepts. Fuzzy Sets and Systems, 147(2), 171-199. doi:10.1016/j.fss.2004.02.009Brown, L. M., Ertürk, R., & Dost, Ş. (2004). Ditopological texture spaces and fuzzy topology, II. Topological considerations. Fuzzy Sets and Systems, 147(2), 201-231. doi:10.1016/j.fss.2004.02.010Brown, L. M., Ertürk, R., & Dost, Ş. (2006). Ditopological texture spaces and fuzzy topology—III: Separation axioms. Fuzzy Sets and Systems, 157(14), 1886-1912. doi:10.1016/j.fss.2006.02.001Diker, M., & Altay Uğur, A. (2012). Textures and covering based rough sets. Information Sciences, 184(1), 44-63. doi:10.1016/j.ins.2011.08.012Dost, Ş. (2017). Semi-compactness in ditopological texture spaces and soft fuzzy topological spaces. Journal of Intelligent & Fuzzy Systems, 32(1), 925-936. doi:10.3233/jifs-1614

Some categorical aspects of the inverse limits in ditopological context

[EN] This paper considers some various categorical aspects of the inverse systems (projective spectrums) and inverse limits described in the category ifPDitop, whose objects are ditopological plain texture spaces and morphisms are bicontinuous point functions satisfying a compatibility condition between those spaces. In this context, the category InvifPDitop consisting of the inverse systems constructed by the objects and morphisms of ifPDitop, besides the inverse systems of mappings, described between inverse systems, is introduced, and the related ideas are studied in a categorical - functorial setting. In conclusion, an identity natural transformation is obtained in the context of inverse systems - limits constructed in ifPDitop and the ditopological infinite products are characterized by the finite products via inverse limits.Yildiz, F. (2018). Some categorical aspects of the inverse limits in ditopological context. Applied General Topology. 19(1):101-127. https://doi.org/10.4995/agt.2018.781210112719

GENERALIZATION SOME FUZZY SEPARATION AXIOMS TO DITOPOLOGICAL TEXTURE SPACES

The authors characterize the notion of quasi coincident in texture spaces and study the generalization of fuzzy quasi separation axioms defined by [12] to the ditopological texture spaces

Generalized closed sets in ditopological texture spaces with application in rough set theory

In this paper, the counterparts of generalized open (g-open) and generalized closed (g-closed) sets for ditopological texture spaces are introduced and some of their characterizations are obtained. Some characterizations are presented for generalized bicontinuous difunctions. Also, we introduce new notions of compactness and stability in ditopological texture spaces based on the notion of g-open and g-closed sets. Finally, as an application of g-open and g-closed sets, we generalize the subsystem based denition of rough set theory by using new subsystem, called generalized open sets to dene new types of lower and upper approximation operators, called g-lower and g-upper approximations. These decrease the upper approximation and increase the lower approximation and hence increase the accuracy. Properties of these approximations are studied. An example of multi-valued information systems are given

Sequentially dinormal ditopological texture spaces and dimetrizability

AbstractThe authors extend the bitopological notion of sequential normality to ditopological texture spaces, and use this notion to state and prove a (pseudo-)dimetrizability theorem

On classes of T0 spaces admitting completions

[EN] For a given class X of T0 spaces the existence of a subclass C, having the same properties that the class of complete metric spaces has in the class of all metric spaces and non-expansive maps, is investigated. A positive example is the class of all T0 spaces, with C the class of sober T0 spaces, and a negative example is the class of Tychonoff spaces. We prove that X has the previous property (i.e., admits completions) whenever it is the class of T0 spaces of an hereditary coreflective subcategory of a suitable supercategory of the category Top of topological spaces. Two classes of examples are provided.Giuli, E. (2003). On classes of T0 spaces admitting completions. Applied General Topology. 4(1):143-155. doi:10.4995/agt.2003.2016.SWORD1431554

Topological Properties of Generalized Context Structures

Práce je zaměřena na vzájemnou interakci několika odvětví matematiky. Hlavní myšlenkou práce bylo najít závislosti, vztahy a analogie mezi nimi. První část práce se týká vztahu mezi formální pojmovou analýzou, topologií a parciálními metrikami. Formální kontext je velice obecná matematická struktura, která může reprezentovat ostatní matematické struktury v jednotné a sjednocené formě. Přirozeným způsobem bychom mohli reprezentovat informaci podobně jako v tabulce, reprezentující formální kontext (s respektem ke všem množinově-teoretickým omezením) a generovat určité topologie na množinách atributů a objektů. V druhé části studujeme především pretopologické systémy jako speciální případ formálních kontextů. Od topologických systémů se pretopologické systémy liší především obecnější uspořádanou strukturou na množině atributů, reprezentujících zobecněné otevřené množiny. Vlastnosti tohoto uspořádání podstatně ovlivňují chování celé struktury a proto mu věnujeme zvláštní pozornost v závěru kapitoly, kde se mj. zabýváme konstrukcí analogie de Grootova duálu, včetně jeho iterovaných vlastností. Třetí část práce je zasvěcena struktuře framework, která má přirozenou strukturu formálního kontextu. Framework se skládá ze dvojice množin, z nichž první je množina míst a druhá obsahuje jistý systém podmnožin první množiny, aniž by bylo vyžadováno splnění nějakých axiómů. Struktura je opatřena jednoduchou konstrukcí duality, umožňující přepínání mezi klasickým, bodově-množinovým přístupem, podobně jako v topologii a bezbodovou reprezentací topologických vztahů. V závěru navrhujeme a studujeme, jak aproximovat libovolný framework pomocí usměrněného souboru konečných frameworků z hlediska generované topologie. V poslední části práce používáme metody obecné topologie ke korekci a zlepšení jednoho ze základních teorémů teorie her. Dokázali jsme mimo jiné, že pro hru v normální formě, v níž má i-tý hráč spojitou výherní funkci a množina jeho strategií je skoro-kompaktní, má tento hráč nedominovanou strategii. Kromě tohoto výsledku v poslední a předposlední kapitole ukazujeme, že teorie her přirozeným způsobem generuje velmi obecné, například nehausdorffovské topologické a kontextové struktury, čímž posouvá tradiční chápání reality neobvyklým směrem.This work is focused on the interaction of several branches of mathematics. The main idea was to nd dependencies, relationships and analogies between them. First part of the work is concerned to the relationship between Formal Concept Analysis, General Topology and Partial Metrics. A formal context is a very general mathematical structure that can represent other mathematical structures in a unied form. In a natural way, we could represent an information in a cross-table-like view of a formal context (fully respecting all set-theoretical limitations) and generate a topology on an attribute and object sets. In the second part the we study especially the pretopological systems as a special case of the formal contexts. They dier from topological systems especially by a more general poset structure of the set of attributes, representing the generalized open sets. Since the properties of this order structure are essential for the behavior of the whole structure, we pay them a special attention at the end of the chapter. Among others, we construct and study an analogue of the de Groot dual for posets, including its iteration properties. The third part is devoted to a mathematical structure called framework that has a contextual nature. A framework consists of two sets, rst one is a set of places, and the second one is a family of some its subsets, without the necessity of any external axioms to be fullled. The structure is equipped with a simple duality construction, allowing to switch between the classical point-set representation (like in topological spaces) and the point-less representation of topological relationships. At the end of the chapter, we suggest and study how a framework could be approximated by a directed family of nite frameworks from the point of view of the generated topology. In the last part the general topology methods were used to correct and improve one of the fundamental theorems in the game theory. It was showed that in a normal form game if i-th player has a continuous utility function and if the set of his strategies is almost-compact then he has an undominated strategy. In addition to this result, in the last two chapters we show that game theory naturally generates very general, for instance non-Hausdor topological and context structures, which shifts the traditional perception of reality in unexpected direction.