SOMEWHAT gp-CONTINUOUS AND SOMEWHAT gp-IRRESOLUTE FUNCTIONS

The notion of fuzzy sets initiated to overcome the uncertainty of an object. Fuzzy topological space, in- tuitionistic fuzzy sets in topological structure space, vagueness in topological structure space, rough sets in topological space, theory of hesitancy and neutrosophic topological space, etc. are the extension of fuzzy sets. Soft set is a family of parameters which is also a set. Fuzzy soft topological space, intuitionistic fuzzy soft and neutrosophic soft topological space are obtained by incorporating soft sets with various topological structures. This motivates to write a review and study on various soft set concepts. This paper shows the detailed review of soft topological spaces in various sets like fuzzy, Intuitionistic fuzzy set and neutrosophy. Eventually, we compared some of the existing tools in the literature for easy understanding and exhibited their advantages and limitations.

This paper explores the decomposition of pre--irresolute (pre--irr) maps and their connection to generalized closed sets (g-closed) in topological space (top. sp.). We provide the necessary background on top. sp., continuity, closed, and related concepts. We introduce the definitions of pre--irr maps and g-closed and discuss their basic properties. The decomposition theorem is presented, stating that every pre--irr map can be decomposed into a composition of a pre--continuous (pre- -cont.) map and a b-irr map. Examples and applications are provided to illustrate the behavior of pre--irr maps and the practical implications of the decomposition theorem. We examine the relationships between pre--cont. maps and pre--irr maps, exploring the conditions under which a pre--irr map is also -irr, and provide a proof for this relationship. The properties of pre--cont. maps, - maps, g- pre--cont. maps, and g--cont. maps are discussed, along with proofs and examples to illustrate their applicability. This paper contributes to the understanding of pre--irr maps, their decomposition, and their relationships with g-closed, providing a foundation for further research in top. sp. and map decompositions. Keywords: pre-β-irresolute maps, g-closed, pre-β-cont. maps, b-cont. maps

In this work we introduce the supra iR-open set in supra topological spaces. , a novel class of supra i-open sets. This class of sets exactly lies between supra regular open and supra i- open sets. We also look at its basic characteristics and compare it to other sorts of data supra semi open sets types as supra i- open set, supra regular open, regular open, regular closed, clopen, supra semi regular open set and supra clopen . By using this set , We introduce and define the notion of supra totally iR-continuous , supra iR-irresolute map and investigate some of its properties. We prove that every supra totally iR- continuous function is supra iR-irresolute, every totally i????-continuous function is supra totally iR continuous and any strongly continuous function is supra totally iR-continuous, on the other hand, we give examples to show that the convers may not be true. At last , We have proved some theorem to discuss the properties of supra iR-normal space and supra iR-regular space.

In this paper we introduce a new type of closed sets called $\varsigma \delta \hat{g}$ closed sets in a topological space $X$, defined in terms of grills. The relations between $\varsigma \delta \hat{g}$ closed sets and already available closed sets are discussed.

In this paper, we show that some results obtained in [S. Sekar and R. Brindha, On Pre Generalized B-Closed Set In Topological Spaces, Int. J. Pure Appl. Math., 111(4)(2016),577-586] are incorrect in general,by giving counter examples. Also, we illustrate that Example 3.5,Example 3.13, Example 3.15 and Example 3.17 are incorrect. Moreover, the correction form of the incorrect results of [13] is presented. Finally, we established that the concepts of b-closed sets and pgb-closed sets are same.

In this paper, we introduce a new concept called semi-perfect sets in topological spaces. We define semi-isolated points and semi-perfect sets. Further, some interesting results and properties of these concepts are investigated.

This paper introduces the notion of generalized hyperconnected sets in topological spaces using Levine’s concept of generalized closed sets. We define generalized hyperconnected sets and generalized hyperconnected components and study their fundamental properties. Furthermore, we provide necessary conditions for a topological space to be generalized hyperconnected. The relationship between this type of connectedness and other types is also presented. Several results concerning these new concepts are obtained and proven.

The concept main of paper is to connected between two definitions pre-open sets and g-open sets in the topological spaces is called Hpre-open sets, study restudy relations them, also present a new definitions, namely Hpre-continous, Hpre-open functions, Hpre-closed functions and have proved some theorems about these concept. Finally, we have present the definitions and results of separation axioms by Hpre-open sets,namely Hpre - T1 spaces , Hpre - T2 spaces and studied some of its applications.

In 2016 A. Devika and A. Thilagavathi introduced a new class of sets called M*-open sets and investigated some properties of these sets in topological spaces. In this paper, we introduce and study a new class of spaces, namely M*-irresolute topological vector spaces via M*-open sets. We explore and investigate several properties and characterizations of this new notion of M*-irresolute topological vector space. We give several characterizations of M*-Hausdorff space. Moreover, we show that the extreme point of the convex subset of M*-irresolute topological vector space X lies on the boundary.

In the current study, the idea of S ψ φ -open sets is used to introduce some new types of nearly open sets in topological space, where ψ = { θ , δ , τ } and φ = { θ , δ , τ , α , a } , thereby illustrating some of their basic qualities and the links between them. Some characterizations, topological properties, mappings, and separation axioms of these new concepts are also presented.

A fixed‐point theorem is proved under noncompact setting of general topological spaces. By applying the fixed‐point theorem, several new existence theorems of solutions for equilibrium problems are proved under noncompact setting of topological spaces. These theorems improve and generalize the corresponding results in related literature.

Existence of solutions for quasi-equilibrium problems in noncompact topological spaces

In this paper, we introduce and study the new class of sets, namely Regular Feebly Generalized Closed (briefly \(RFG\)-closed) sets, Regular Feebly Generalized neighborhoods (briefly \(RFG\)-nbhd), \(RFG\)-interior and \(RFG\)-closure in topological spaces and also some properties of new concepts have be en studied.

In this paper, a new class of called semi-pre-generalised closed set in topological space are introduced and studied. Also some of their properties have been investigated. We also introduce sp-T1/2 space, and studied the relationship between spg-closedset and some space in topological space, as well discuss some properties, theorem, corollary and examples.

In the year 2005, the authors introduced and studied the concept of pgpr-closed sets in topological spaces. The purpose of this paper is to introduce locally and weakly pgpr-closed sets and investigate their basic properties.