Relative compactness, cotopology and some other notions from the bitopological point of view

AbstractThe paper consists of two sections. Section 1 is the introduction which, in addition to the auxiliary information, contains some interesting results on Baire-like properties. Section 2 deals with the bitopological essence of the notions of relative compactness and cotopology in general topology, C-relation, subordination of topologies and closed neighborhoods condition in analysis. A generalization of Choquet's theorem on Baire spaces is given and the sufficient conditions for families of (i,j)-nowhere dense sets to coincide with families of (i,j)-first category sets are established using a finite measure. A bitopological solution of one of Ulam's problems is obtained. The corresponding relations are almost always studied using essentially the bitopological modifications of regularity, which, as seen in various problems of general topology, analysis and potential theory, are the most natural forms of relations of two topologies defined on the same set

Some Types of Mappings in Bitopological Spaces

            قدمنا بعض المفاهيم في الفضاءات التبولوجية الثنائية وهي الاقتراب من المجموعة الجزئية من النمط nm-j-ω ، الاتجاه المباشر لمجموعة من النمط nm- j-ω ، التطبيقات المغلقة من النمط nm- j-ω ، صلابة المجموعة من النمط nm- j-ω ، التطبيقات المستمرة من النمط nm- j-ω ، والخط الرئيسي لهذا البحث هو التطبيقات التامة من النمط nm- j-ω في الفضاءات التبولوجية الثنائية. المميزات المتعلقة بهذه المفاهيم والعديد من المبرهنات قد درسنا حيث j = q , δ, a , pre, b, b.            This work, introduces some concepts in bitopological spaces, which are nm-j-ω-converges to a subset, nm-j-ω-directed toward a set, nm-j-ω-closed mappings, nm-j-ω-rigid set, and nm-j-ω-continuous mappings. The mainline idea in this paper is nm-j-ω-perfect mappings in bitopological spaces such that n = 1,2  and m =1,2 n ≠ m. Characterizations concerning these concepts and several theorems are studied, where j = q , δ, a , pre, b, b.

Near ω-continuous multifunctions on bitopological spaces

In this paper, we introduce and study basic characterizations, several properties of upper (lower) nearly (i; j)-!-continuous multifunctions on bitopological space

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

Pointfree bispaces and pointfree bisubspaces

This thesis is concerned with the study of pointfree bispaces, and in particular with the pointfree notion of inclusion of bisubspaces. We mostly work in the context of d-frames. We study quotients of d-frames as pointfree analogues of the topological notion of bisubspace. We show that for every d-frame L there is a d-frame A(L) such that it plays the role of the assembly of a frame, in the sense that it has the analogue of the universal property of the assembly and that its spectrum is a bitopological version of the Skula space of the bispace dpt(L), the spectrum of L. Furthermore, we show that this bitopological version of the Skula space of dpt(L) is the coarsest topology in which the d-sober bisubspaces of dpt(L) are closed. We also show that there are two free constructions in the category of d-frames Act(L) and A_(L), such that they represent two variations of the bitopological version of the Skula topology. In particular, we show that in dpt(Act) the positive closed sets are exactly those d-sober subspaces of dpt(L) that are spectra of quotients coming from an increase in the con component, and that the negative closed ones are those that come from increases in the tot component. For dpt(A_(L)), we show that the positive closed sets are exactly those bisubspaces of dpt(L) that are spectra of quotients coming from a quotient of L+, and that the negative closed sets come in the same way from quotients of

On Contra gy-Continuous Functions

In this paper, we investigate further properties of the notion of contra gy-continuous functions which was introduced in [4]. We obtain some separation axioms of contra gy-continuous functions and discuss the relationships between contra gy-continuity and other related functions