On generalized γµ-closed sets and related continuity

In this paper our main interest is to introduce a new type of generalized open sets defined in terms of an operation on a generalized topological space. We have studied some properties of this newly defined sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have studied some preservation theorems in terms of some irresolute functions.

Uniformity on generalized topological spaces

PurposeThe present article deals with the initiation and study of a uniformity like notion, captioned μ-uniformity, in the context of a generalized topological space.Design/methodology/approachThe existence of uniformity for a completely regular topological space is well-known, and the interrelation of this structure with a proximity is also well-studied. Using this idea, a structure on generalized topological space has been developed, to establish the same type of compatibility in the corresponding frameworks.FindingsIt is proved, among other things, that a μ-uniformity on a non-empty set X always induces a generalized topology on X, which is μ-completely regular too. In the last theorem of the paper, the authors develop a relation between μ-proximity and μ-uniformity by showing that every μ-uniformity generates a μ-proximity, both giving the same generalized topology on the underlying set.Originality/valueIt is an original work influenced by the previous works that have been done on generalized topological spaces. A kind of generalization has been done in this article, that has produced an intermediate structure to the already known generalized topological spaces.

Generalized Topological Groupoids

Our aim in this paper is to give the notion of generalized topological groupoid which is a generalization of the topological groupoid by using the notion of generalized topology defined by Csasz ´ ar [6]. We in- ´ vestigate the basic facts in the groupoid theory in terms of generalized topological groupoids. We present the action of a generalized topological groupoid on a generalized topological space. We obtain some characterizations about this concept that is called the generalized topological action. Beside these, we give definition of a generalized topological crossed module by generalizing the concept of crossed module defined on topological groupoids. At the last part of the study, we show how a generalized topological crossed module can be obtained from a generalized topological groupoid and how a generalized topological groupoid can be obtained from a generalized topological crossed module.

Generalized topologies with associating function and logical applications

The whole universe of a generalized topological space may not be open. Hence, some points may be beyond any open set. In this paper we assume that such points are associated with certain open neighbourhoods by means of a special function F. We study various properties of the structures obtained in this way. We introduce the notions of F-interior and F-closure and we discuss issues of convergence in this new setting. It is possible to treat our spaces as a semantical framework for modal logic.

Applications on operations on weakly compact generalized topological spaces

In this paper, we have introduced the notion of operations on a generalized topological space $(X,\mu)$ to investigate the notion of $\gamma_{_\mu}$-compact subsets of a generalized topological space and to study some of its properties. It is also shown that, under some conditions, $\gamma_{_\mu}$-compactness of a space is equivalent to some other weak forms of compactness. Characterizations of such sets are given. We have then introduced the concept of $\gamma_{_\mu}$-$T_{_2}$ spaces to study some properties of $\gamma_{_\mu}$-compact spaces. This operation enables us to unify different results due to S. Kasahara.

Theory of g-Tg-Derived and g-Tg-Coderived Operators: Definitions, Essential Properties, Iterations, and Ranks

In a generalized topological space Tg = (Ω, Tg) (Tg-space), the g-topology Tg : P (Ω) −→ P (Ω) can be characterized in the generalized sense by specifying the generalized open, generalized closed sets (g-Tg-open, g-Tg-closed sets), generalized interior, generalized closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω) (g-Tg-interior, g-Tg-closure operators), or generalized derived, generalized coderived operators g-Derg, g-Codg : P (Ω) −→ P (Ω) (g-Tg-derived, g-Tg-coderived operators), respectively. For very many Tg-spaces, the δth-iterates g-Derg(δ), g-Codg(δ) : P (Ω) −→ P (Ω) of g-Derg, g-Codg : P (Ω) −→ P (Ω), respectively, defined by transfinite recursion on the class of successor ordinals are also themselves g-Tg-derived, g-Tg-coderived operators for new g-topologies in the generalized sense on Ω. Thus, the use of novel definitions of g-Tg-derived, g-Tg-coderived operators g-Derg, g-Codg : P (Ω) −→ P (Ω), respectively, based on a very clever construction, together with their δth-iterates g-Tg-operators g-Derg(δ), g-Codg(δ) : P (Ω) −→ P (Ω), defined by transfinite recursion on the class of successor ordinals, will give rise to novel generalized g-topologies on Ω. The present authors have been actively engaged in the study of g-Tg-operators in Tg-spaces. The study of the essential properties and the commutativity of novel definitions of g-Tg-interior and g-Tg-closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω), respectively, in Tg has formed the first part, and the study of the essential properties and sets of consistent, independent axioms of novel definitions of g-Tg-exterior and g-Tg-frontier operators g-Extg, g-Frg : P (Ω) −→ P (Ω), respectively, has formed the second part. In this work, which forms the last part on the theory of g-Tg-operators in Tg-spaces, the present authors propose to present novel definitions and the study of the essential properties of g-Tg-derived and g-Tg-coderived operators g-Derg, g-Codg : P (Ω) −→ P (Ω), respectively, and their δth-iterates, and the notions of g-Tg-open and g-Tg-closed sets of ranks δ in Tg-spaces.

On functions continuous with respect to a density type strong generalized topology

In the paper, some properties of functions continuous with respect to a density type strong generalized topology are presented. In particular, it is proved that each real function is approximately continuous with respect to this generalized topology almost everywhere. Moreover, some separation axioms for this generalized topological space are investigated.

On (μ1, μ2, μ3)-Weakly Generalized Closed Sets

This paper defines a new generalization of closed sets in a tri-generalized topological space called (μ1, μ2, μ3)-weakly generalized closed set (or briefly (μ1,μ2, μ3)-wg closed set) which is defined as follows: A subset A of X is (μ1, μ2, μ3)-weakly generalized closed set if clμ1(intμ2(A)) ⊆U whenever A ⊆ U and U is μ3-open in X. At least fifteen defined closed sets found in literature are considered special cases of (μ1, μ2, μ3)-weakly generalized closed set under some conditions. Furthermore, some properties of (μ1, μ2, μ3)-weakly generalized closed sets are obtained.