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Journal articles on the topic 'And beeta*g-open sets'

Author: Grafiati

Published: 7 June 2025

Last updated: 11 July 2025

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1

Hamant, Kumar Hamant. "On beeta*g-closed sets and beeta*-normal spaces." Acta Ciencia Indica XLI M, no. 1 (2025): 67–72. https://doi.org/10.5281/zenodo.14866811.

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In this paper, we introduce the notion of beeta*g-closed sets and we show that the family of all beeta*g-open sets in a topological spaces (X, T) is a topology for X which is finer than T. Further we obtain some characterizations and preservation theorems for beeta*-normality and normality. 

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2

Hamant, Kumar Hamant. "Quasi beeta-normal spaces and pi g beeta-closed functions." Acta Ciencia Indica XXXVIII M, no. 1 (2025): 149–54. https://doi.org/10.5281/zenodo.14866719.

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In this paper, by using pi g beeta-closed sets, we introduce some functions such as pi g beeta-closed, almost pigbeeta-closed, pigbeeta-continuous and almost pigbeeta-continuous functions. Further we obtain a characterization and preservation theorems for quasi beeta-normal spaces.

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3

Hamant, Kumar Hamant. "pigbeeta-Normal Spaces in Topological Spaces." International Journal of Science and Research (IJSR) 4, no. 2 (2025): 1531–34. https://doi.org/10.5281/zenodo.14866928.

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The aim of this paper is to introduce a new class of normal spaces called pigbeeta-normal spaces, by using pigbeeta-open  sets. We prove that pigbeeta-normality is a topological property and it is a hereditary property with respect to  pi-open, pigbeeta-closed subspaces. Further we obtain a characterization and preservation theorems for g-normal spaces. 

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4

Hamant, Kumar Hamant. "etag-closed Sets and eta-Normal Spaces in Topological Spaces." International Journal of Creative Research Thoughts (IJCRT) 10, no. 3 (2022): 1–7. https://doi.org/10.5281/zenodo.14880864.

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The aim of this paper is to introduce and study a new class of sets called etag-closed sets and a new class of spaces called eta-normal spaces. The relationships among beeta*g-normal, alpha-normal, s-normal and eta-normal spaces are investigated. Moreover, we introduce the concept of eta-generalized closed functions. We also obtain some characterizations and preservation theorems of eta-normal spaces, in the forms of generalized eta-closed and eta-generalized closed functions.

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5

Hamant, Kumar Hamant. "β*g-normal spaces in topological spaces". Journal of Emerging Technologies and Innovative Research (JETIR) 6, № 6 (2019): 897–907. https://doi.org/10.5281/zenodo.14885821.

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The aim of this paper is to introduce and study a new class of spaces, called β*g-normal spaces. The relationships among s-normal spaces, p-normal spaces, alpha-normal spaces, β-normal spaces, gamma-normal spaces and β*g-normal spaces are investigated. Moreover, we introduce the forms of generalized β*g-closed and β*g-generalized closed functions. We obtain characterizations of β*g-normal spaces, properties of the forms of generalized β*g-closed functions and preservation theorems.

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6

admin, admin, and G. Chandra Ray. "A new class of NeutroOpen, NeutroClosed, AntiOpen and AntiClosed sets in NeutroTopological and AntiTopological spaces." International Journal of Neutrosophic Science 20, no. 2 (2023): 77–85. http://dx.doi.org/10.54216/ijns.200206.

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A lot of research has been done on the types of open and closed sets in general topological spaces and also in general bitopological spaces. Types of sets like pre-open sets and pre-closed sets, semi-open sets and semi-closed sets, Alpha-open sets, and Alpha-closed sets, regular open sets and regular closed sets, g-open sets and g-closed sets, and many more have been defined and studied. In the current study, an attempt has been made to define and give examples of a new category of open and closed sets, namely, NeutroOpen and NeutroClosed sets. Further, the concept of neutron-topology is used to define NeutroPreOpen and NeutroPreClosed sets, NeutroSemiOpen and NeutroSemiClosed sets, NeutroAlphaOpen and NeutroAlphaClosed sets, NeutroRegularOpen and NeutroRegularClosed sets, NeutroBetaOpen, and NeutroBetaClosed sets, and several examples have been given to illustrate each of the new classes of sets. Also, the concept of AntiTopology has been used to define another class of sets, namely, AntiOpen and AntiClosed sets of the above five classes of sets, namely, regular-openclosed; semi-openclosed, Alpha-openclosed, Beta-openclosed pre-openclosed sets. Further, a new class of subsets is identified which are named as NeutroTauOpen and NeutroTauClosed sets. Similar subsets in anti-topological spaces are named as AntiTauOpen and AntiTauClosed sets.

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7

Hamant, Kumar Hamant. "II-NORMAL SPACES IN TOPOLOGICAL SPACES." Journal of Emerging Technologies and Innovative Research (JETIR) 7, no. 8 (2020): 1510–20. https://doi.org/10.5281/zenodo.14885784.

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The aim of this paper is to introduce and study a new class of spaces, called ii-normal spaces. The relationships among beeta*g-normal, s-normal, alpha-normal, gamma-normal and ii-normal spaces are investigated. Moreover, we introduce the forms of generalized ii-closed (briefly gii-closed) and ii-generalized closed (briefly iig-closed) functions. We obtain characterizations of ii-normal spaces, properties of the forms of generalized ii-closed functions and preservation theorems.

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8

Taha, Islam, Jawaher Al-Mufarrij, and Osama Taha. "Some Characterizations of \((r,s)\)-Fuzzy \(b\)-Open Sets with Applications in Double Fuzzy Topological Spaces." European Journal of Pure and Applied Mathematics 18, no. 2 (2025): 5911. https://doi.org/10.29020/nybg.ejpam.v18i2.5911.

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In this paper, we displayed and characterized a novel class of fuzzy open sets ($\mathcal{F}$-open sets) in double fuzzy topological spaces ($\mathcal{DFTS}s$) based on \v{S}ostak$^{,}$s sense, called $(r,s)$-fuzzy $b$-open sets ($(r,s)$-$\mathcal{F}$-$b$-open sets). This class is contained in the class of $(r,s)$-$\mathcal{F}$-$\beta$-open sets and contains all $(r,s)$-$\mathcal{F}$-$\alpha$-open sets, $(r,s)$-$\mathcal{F}$-pre-open sets, and $(r,s)$-$\mathcal{F}$-semi-open sets. Next, we explored and studied the notion of $\mathcal{DF}$-$b$-continuity between $\mathcal{DFTS}s$ $(G, \Im, \Im^*)$ and $(Z, \digamma, \digamma^*)$. We also defined and discussed the notions of $\mathcal{DF}$-almost $b$-continuity and $\mathcal{DF}$-weakly $b$-continuity, which are weaker forms of $\mathcal{DF}$-$b$-continuity. Thereafter, we presented and investigated novel $\mathcal{DF}$-mappings via $(r,s)$-$\mathcal{F}$-$b$-open and $(r,s)$-$\mathcal{F}$-$b$-closed sets. Finally, we introduced some novel types of $\mathcal{DF}$-separation axioms, called $(r,s)$-$\mathcal{F}$-$b$-regular and $(r,s)$-$\mathcal{F}$-$b$-normal spaces, and studied some properties of them.

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9

Powar, P. L., Baravan Asaad, K. Rajak, and R. Kushwaha. "Operation on Fine Topology." European Journal of Pure and Applied Mathematics 12, no. 3 (2019): 960–77. http://dx.doi.org/10.29020/nybg.ejpam.v12i3.3449.

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This paper introduces the concept of an operation $\gamma$ on $\tau_f$. Using this operation, we define the concept of $f_\gamma$-open sets, and study some of their related notions. Also, we introduce the concept of $f_\gamma g$.closed sets and then study some of its properties. Moreover, we introduce and investigate some types of $f_\gamma$-separation axioms and $f_{\gamma\beta}$-continuous functions by utilizing the operation $\gamma$ on $\tau_f$. Finally, some basic properties of functions with $f_\beta$-closed graphs have been obtained.

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10

C., Sivashanmugaraja, та Antony Jeroldz S. "Fuzzy β-γ-open sets and fuzzy β-γ-open mappings". Applied Science and Computer Mathematics 2, № 2 (2022): 66–71. https://doi.org/10.5281/zenodo.6298753.

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11

I., Rajasekaran. "Weak forms of strongly nano open sets in ideal nanotopological spaces." Asia Mathematika 5, no. 2 (2021): 96–102. https://doi.org/10.5281/zenodo.5518562.

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The object of the paperwork is to define new concepts called strongly \(\alpha-nI\)-open sets, strongly pre-nI-open sets, strongly b-nI-open sets, and strongly \(\beta-nI\)-open sets, which are weak forms of nano open sets in ideal nano topological spaces. Also, we characterize the relations between them and the related properties. 

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12

Selvaraj, Ganesan, and Smarandache Florentin. "Some new classes of neutrosophic minimal open sets." Asia Mathematika 5, no. 1 (2021): 103–12. https://doi.org/10.5281/zenodo.4724804.

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This article focuses on \(N_m- \beta\)-open, \(\beta\)-interior and \(\beta\)-closure operators using neutrosophic minimal structures. We investigate the properties of such concepts and we introduced the concepts of \(N_m- \beta\)-continuous,  \(N_m- \beta\)-closed graph, \(N_m- \beta\)-compact and almost \(N_m- \beta\)-compact. Finally, we introduced the concepts of \(N_m-\)regular-open sets and \(N_m-\pi\)-open sets and investigate some properties.

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13

AL-Jarrah, Heyam H., Amani Rawshdeh, Khalid Y. Al-Zoubi, and Shefa A. Bani Melhem. "On $g \mu$-Paracompact Sets." European Journal of Pure and Applied Mathematics 18, no. 2 (2025): 5899. https://doi.org/10.29020/nybg.ejpam.v18i2.5899.

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In this work, we use the notion of the $g \mu$-paracompact space \cite{ADeb} to introduce two types of $g \mu$-paracompact sets called $\alpha$-$g \mu$-paracompact and $\beta$-$g \mu$-paracompact. We show that every $\alpha$-$g \mu$-paracompact set is $\beta$-$g \mu$-paracompact and if a generalized topological space $(S,\mu)$ is $g \mu$-paracompact, then every $\mu$-closed subset in $(S,\mu)$ is $\alpha$-$g \mu$-paracompact while every $\mu g$-closed subset in $(S, \mu)$ is $\beta$-$g \mu$-paracompact. Finally, we introduce the notion of $co$-$\alpha$-$g\mu$-paracompact set as an application of $\alpha$-$g\mu$-paracompact and study some of its features.

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14

Journal, Baghdad Science. "ON G-OPEN SET." Baghdad Science Journal 4, no. 3 (2007): 482–84. http://dx.doi.org/10.21123/bsj.4.3.482-484.

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A new class of generalized open sets in a topological space, called G-open sets, is introduced and studied. This class contains all semi-open, preopen, b-open and semi-preopen sets. It is proved that the topology generated by G-open sets contains the topology generated by preopen,b-open and semi-preopen sets respectively.

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15

Shakir, Qays Rashid. "Minimal and Maximal Beta Open Sets." Journal of Al-Nahrain University Science 17, no. 1 (2017): 160–66. http://dx.doi.org/10.22401/jnus.17.1.22.

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16

FETHULLAH, EROL, and OZKAN ALKAN. "SOFT Beta-OPEN SET VIA SOFT IDEALS." Asian Journal of Current Research 2, no. 1 (2017): 10–19. https://doi.org/10.5281/zenodo.1408476.

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In this paper, we introduced the notions of soft  Beta I I sets and soft  Beta I I continuous functions. Also we obtain some characterizations and several properties concerning these functions.

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17

A., S. Nawar, та M. Atallah H. "On δβ -Open Sets in Tri Topological Spaces". Journal of Advanced Studies in Topology 12, № 1-2 (2022): 8–17. https://doi.org/10.5281/zenodo.7330008.

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The main aim of this paper is to study δβ-open sets in tri topological spaces along with their several properties and characterizations. δβδβ-continuous and δβδβ-irresolute functions and some of their basic properties are discussed. Some new spaces in tri topological spaces, called τ123-δβ-Tk spaces, k=0,1,2 are introduced and their properties and characterizations are analyzed.

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18

Dodamani, Abeda, and Leonard Winston Aiman. "Soft \(\beta^*\) open sets in soft topological spaces." Malaya Journal of Matematik 13, no. 01 (2025): 46–53. https://doi.org/10.26637/mjm1301/006.

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In this present paper, we define the concepts of soft \(\beta^*\) -open sets, soft \(\beta^*\) -closed sets in soft topological space and study some of their properties. Further, the notion of soft \(\beta^*\) interior and soft \(\beta^*\) closure of a set are introduced, and their basic properties are explored.

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19

Fayad, Eleyan, and Hisham Mahdi. "Soft \beta c-open sets and soft \beta c-continuity." International Mathematical Forum 12 (2017): 9–26. http://dx.doi.org/10.12988/imf.2017.611150.

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20

NETHAJI, OCHANAN, SUBRAMANIYAN DEVI, MAYANDI RAMESHPANDI, and RAJENDARAN PREMKUMAR. "DECOMPOSITIONS OF NANO g^#-CONTINUITY VIA IDEALIZATION." Journal of Science and Arts 23, no. 2 (2023): 377–80. http://dx.doi.org/10.46939/j.sci.arts-23.2-a05.

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In this paper we introduce the notions of αg^#-nI-open sets, η^#-nI-open sets, h^#-nI-open sets, g^# _t-nI-sets, g^# _(α^* )-nI-sets and g^# _S-nI-sets in ideal nano topological spaces and investigate some of their properties.

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21

Caldas, Miguel. "Certain types of functions via $\boldsymbol{\beta\theta}$-open sets." Sarajevo Journal of Mathematics 13, no. 1 (2017): 105–13. http://dx.doi.org/10.5644/sjm.13.1.08.

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New classes of functions, called somewhat $\beta\theta$-continuous, somewhat $\beta\theta$-open and hardly $\beta\theta$-open functions, has been defined and studied by making use of $\beta\theta$-open sets. Characterizations and properties of these functions are presented.

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22

Alsharari, Fahad, and Abdo Qahis. "On pairwise \(\mathcal{N}\)-\(\beta\)-open sets in bitopological spaces." Journal of Advanced Studies in Topology 7, no. 3 (2016): 152. http://dx.doi.org/10.20454/jast.2016.1099.

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In this paper, we introduce the notion of an \((i,j)\)-\(\mathcal{N}\)-\(\beta\)-open set which is a generalization of an \((i,j)\)-\(\beta\)-open set in a bitopological space. Also, we investigate some of its properties and characterizations. Besides, we prove that a pairwise \((i, j)\)-\(\mathcal{N}\)-\(\beta\)-open cover that has a finite (countable) subcover is equivalent to a pairwise \(\beta\)-compact (\(\beta\)-Lindel\"{ö}f) space. Finally, we introduce an \((i, j)\)-\(\mathcal{N}\)-\(\beta\)-continuous function and an \((i, j)\)-\(\mathcal{N}\)-\(\beta\)-irresolute function and obtain some of their properties.

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23

Hassan, Javier, Nuruddina M. Bakar, Norwajir S. Dagsaan, Mercedita A. Langamin, Nurijam Hanna M. Mohammad, and Sisteta U. Kamdon. "J-Open Independent Sets in Graphs." European Journal of Pure and Applied Mathematics 17, no. 2 (2024): 922–30. http://dx.doi.org/10.29020/nybg.ejpam.v17i2.5084.

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Let $G\neq \overline{K}_n$ be a graph with vertex and edge-sets $V(G)$ and $E(G)$, respectively. Then\linebreak $O$ $\subseteq$ $V(G)$ is called a J-open independent set of $G$ if for every $a,b \in V(G)$ where $a\neq b$, $d_G(a,b)$ $\neq 1$, and $N_G(a) \backslash N_G(b) \neq \varnothing$ and $N_G(b)\backslash N_G(a) \neq \varnothing$. The maximum cardinality of a J-open independent set of G, denoted by $\alpha_J(G)$, is called the J-open independence number of $G$. In this paper, we introduce new independence parameter called J-open independence. We show that this parameter is always less than or equal to the standard independence (resp. J-total domination) parameter of a graph. In fact, their differences can be made arbitrarily large. In addition, we show that J-open independence parameter is incomparable with hop independence parameter. Moreover, we derive some formulas and bounds of the parameter for some classes of graphs and the join of two graphs.

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24

Modak, Shyamapada, and Takashi Noiri. "\mu - k - Connectedness in GTS." Boletim da Sociedade Paranaense de Matemática 33, no. 2 (2014): 161–65. http://dx.doi.org/10.5269/bspm.v33i2.23419.

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Csaszar [4] introduced \mu - semi - open sets, \mu - preopen sets, \mu - \alpha - open sets and \mu - \beta - open sets in a GTS (X, \tau). By using the \mu - \sigma - closure, \mu - \pi - closure, \mu - \alpha - closure and \mu - \beta - closure in (X, \tau), we introduce and investigate the notions \mu - k - separated sets and \mu - k - connected sets in (X, \tau).

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25

Asia, Mathematika, and Tahiliani S. "On weakly beta-gamma-regular and weakly beta-gamma-normal spaces." Asia Mathematika 7, no. 3 (2024): 46——54. https://doi.org/10.5281/zenodo.10609732.

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In this paper, we introduce, investigate and study two new topological spaces namely weakly beta-gamma-regular and weakly beta-gamma-normal spaces, by using the concept of operational approach on beta-open sets. The concept of beta-gamma-open sets was introduced, and also we utilized beta-gamma-open sets on beta-regular and beta-normal spaces to study and investigate these new spaces. Some topological properties of weakly beta-gamma-regular and weakly beta-gamma-normal spaces were also studied. Some hereditary properties and preservation theorems are also being obtained.

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26

Powar, P. L., J. K. Maitra та Ramratan Kushwaha. "fγ -PSg-closed sets in Fine Topological Spaces". YMER Digital 21, № 08 (2022): 181–98. http://dx.doi.org/10.37896/ymer21.08/19.

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In this paper, we have defined a new class of sets called fγ-PS-generalized closed sets using fγ-PS-open set and fγ-PS-closure of a set in fine topological space. Also, we have defined some new functions, namely fγ-PS-g-continuous by using the fγ -PS-g-closed set and fγ-PS-g-open set in fine topological space with illustrative examples. AMS Subject Classification: 54XX, 54CXX. Key Words: Fine-open sets, fγ-PS-open sets, fγ-PS-closed sets, fγ-PS-g-continuous function.

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27

Balasubramaniyan, K., S. Sriram, and O. Ravi. "On \(\beta^*g\)-Closed Sets in Fuzzy Topological Spaces." Journal of Advanced Studies in Topology 4, no. 3 (2013): 1. http://dx.doi.org/10.20454/jast.2013.553.

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28

Hussain, Sabir, Bashir Ahmad та Takashi Noiri. "γ*-SEMI-OPEN SETS IN TOPOLOGICAL SPACES". Asian-European Journal of Mathematics 03, № 03 (2010): 427–33. http://dx.doi.org/10.1142/s1793557110000337.

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The concept of generalized open sets in generalized topological spaces was introduced by A. Csaszar [6,7]. In this paper, we introduce the concept of γ*-semi-open sets and investigate the related topological properties of the associated topologies SOγ*(X) and τγ. We also introduce the concepts of [Formula: see text]-sets and Λsγ-sets which generalize g· Λs-sets and g· νs-sets respectively introduced by Minguel Caldas and Julian Dontchev [5].

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29

Saif, Amin, A. Mahdi, and Khaled M. Hamadi. "G-Open sets in Grill Topological Spaces." International Journal of Computer Applications 183, no. 13 (2021): 1–5. http://dx.doi.org/10.5120/ijca2021921446.

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30

Iguda, Abdul. "Compactness With Respect to Ideal Via Beta-Open Sets." Science Forum (Journal of Pure and Applied Sciences) 22, no. 2 (2022): 336. http://dx.doi.org/10.5455/sf.106249.

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31

Altoumi, Nadiy A., and Fatma A. Toumi. "On P^* g- Closed Set in topological Spaces." International Science and Technology Journal 36, no. 1 (2025): 1–10. https://doi.org/10.62341/nafa3003.

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Closed sets play a fundamental role in topological spaces. Notably, a topology on a set can even be characterized by specifying the properties of its closed sets. In 1970, N. Levine introduced the concept of generalized closed sets, defined: A subset S of a topological space X is considered generalized closed if the closure of A is contained in U, cl(A)⊆U whenever A⊆U and U is open set. In this study, we define and explore novel classes of sets termed pre star generalized closed sets (P^* g-closed), pre star generalized open sets (P^* g-open) within the context of topological spaces. The relationship of this set with other closed sets has been proven, and its fundamental properties such as union, intersection, and containment have been established. We have also demonstrated that pre star generalized closed set (P*g- closed) in any given set (let it be X) remains a pre star generalized closed set (P*g-closed) in any of its subset Y⊆X. Moreover, we have proven the fundamental properties of pre star generalized open set (P*g –open). In future studies, we aim to extend this research to introduce a new operator similar to the one currently under investigation, sharing its topological characteristics. Keywords: P-closet, P-open, P^* g-closed, 〖 P〗^* g-open.

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32

Basker, P., та Broumi Said. "On (βρn)-OS in Pythagorean Neutrosophic Topological Spaces". International Journal of Neutrosophic Science 18, № 4 (2022): 183–91. http://dx.doi.org/10.54216/ijns.180417.

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In this paper, we introduce a new set called Pythagorean neutrosophic beta-open set with this concept, and we introduce interior and closure of Pythagorean neutrosophic beta-open set in a Pythagorean neutrosophic topological spaces by utilizing beta-open set and we introduce the chii, ii, iiibeta-spaces and di, ii, iiibetaspaces from the pair of distinct points and we have derived the necessary and sufficient conditions by utilizing beta-open sets. We also go through some containment relations for interiors and closures of beta-open sets and studied some of their characteristics.

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33

Rajakumar, S., A. Vadivel, and K. Vairamanickam. "Minimal \(\tau^*\)-\(g\)-Open Sets and Maximal \(\tau^*\)-\(g\)-Closed Sets in Topological Spaces." Journal of Advanced Studies in Topology 3, no. 3 (2012): 48. http://dx.doi.org/10.20454/jast.2012.195.

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34

S.L.Rathika. "Study on ΛδS−sets I". Proyecciones (Antofagasta) 43, № 6 (2024): 1331–46. https://doi.org/10.22199/issn.0717-6279-5807.

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Several authors started to extend the concept of δ-open sets via var-ious types of generalizations. In 1997, Park et al. introduced a weaker form of δ-open sets called δ-semiopen sets stronger than semiopensets. Maki [6] initiated the notion Λ-sets in topological spaces as theintersection of all open supersets of R. Georgiou et al. [4] definedΛδ-sets, (Λ,δ)-closed and studied. The main aim of this paper is tointroduce an operator ΛδS using δ-semiopen sets and the concept of (Λ,δS)-closed sets in topological spaces. The notions ΛδS(R), ΛδS-sets, Λ∗ δS(R), Λ∗δS-set, (Λ,δS)-closed sets and λδS g-closed sets are de-fined and their properties are studied. It is proved that λδS g-closedsets are weaker than δ-open and δ-semi open sets but stronger thangδS-open sets, δgs-open sets.

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35

K. K, Bushra Beevi, and Baby Chacko. "PARACOMPACTNESS IN GENERALIZED TOPOLOGICAL SPACES." South East Asian J. of Mathematics and Mathematical Sciences 19, no. 01 (2023): 287–300. http://dx.doi.org/10.56827/seajmms.2023.1901.24.

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In this paper we introduce the concepts G - locally finite, σG - locally finite and G - paracompactness. Also discuss about some properties of these concepts. Here we investigate that some properties in topological spaces and generalized topological spaces (GTS) are coincides if we replace open sets by generalized open sets (G - open sets ). Also, we provide some examples to show some results are invalid in the case of GTS.

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36

Sarsak, Mohammad S. "More properties of generalized open sets in generalized topological spaces." Demonstratio Mathematica 55, no. 1 (2022): 404–15. http://dx.doi.org/10.1515/dema-2022-0027.

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Abstract Sarsak [M. S. Sarsak, On some properties of generalized open sets in generalized topological spaces, Demonstr. Math. 46 (2013), no. 2, 415–427] studied some properties of generalized open sets in generalized topological spaces (GTSs); the primary purpose of this article is to investigate more properties of generalized open sets in GTSs. We mainly study the behaviours of regular closed sets, semi-open sets, regular semi-open sets, preopen sets, and β \beta -open sets in GTSs analogous to their behaviours in topological spaces.

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37

Cahis, Abdo Mohammed, та Awn Alqahtani. "Modifications to Mixed θ(ν1, ν2)-open Sets inGeneralized Topological Spaces". European Journal of Pure and Applied Mathematics 17, № 4 (2024): 3610–21. http://dx.doi.org/10.29020/nybg.ejpam.v17i4.5475.

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A. Cs´asz´ar and Makai Jr. [5] introduced the concepts of the mixed operation γθ(ν1,ν2) and mixed θ(ν1, ν2)-open sets in generalized topological spaces. In this paper, we extend this framework by introducing the concepts of mixed operation γ˜θ(ν1,ν2) and mixed ˜θ(ν1, ν2)-open sets (briefly, ˜θ(ν1, ν2)- open sets) and investigate their fundamental properties in generalized topological spaces. We explore the relationships among γ˜θ(ν1,ν2), γθ(ν1,ν2), and γθ(ν), as well as the relationships among ˜θ(ν1, ν2)-open sets, θ(ν1, ν2)-open sets, and μ-open sets. Additionally, we introduce the notion of G(ν1, ν2)-regularity in generalized topological spaces. Finally, we provide characterizations of ˜θ(ν1, ν2)-open sets using mixed G(ν1, ν)-regular concept

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P.R, Kavitha. "NEW SEPARATION AXIOMS VIA *GENERALIZED PRE OPEN SETS." Kongunadu Research Journal 1, no. 2 (2014): 16–19. http://dx.doi.org/10.26524/krj31.

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In this Paper, we introduce the notion of *g-p open sets and *g-p continuity in topological spaces. By tilizing these notions we introduce some weak separation axioms. Also we show that some basic properties of (*g, p)-Ti, (*g, p)-Di spaces, for i = 0, 1, 2,…

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S. Gowri. "On Beta Generalized Star Pre-I-Closed Sets in Ideal Topological Spaces." Communications on Applied Nonlinear Analysis 31, no. 6s (2024): 207–15. http://dx.doi.org/10.52783/cana.v31.1179.

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The purpose of this paper is to define the new idea Beta generalized star pre-closed sets, a new class of closed and open sets in topological spaces, and to examine some of its characteristics using few examples. In addition, we define Beta generalized star pre-I-closed sets, a new class of closed and open sets in ideal topological spaces and discuss through their characteristics.

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Nazir, Ahmad Ahengar*1 &. J.K. Maitra2. "ON g-BINARY m-OPEN SETS AND MAPS." GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES 5, no. 9 (2018): 318–22. https://doi.org/10.5281/zenodo.1442727.

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Latif, Raja Mohammad. "Beta- Star- Continuity and Beta- Star- Contra- Continuity." International Journal of Pure Mathematics 7 (February 8, 2021): 43–66. http://dx.doi.org/10.46300/91019.2020.7.6.

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In 2014 Mubarki, Al-Rshudi, and Al- Juhani introduced and studied the notion of a set in general topology called β*-open set and investigated its fundamental properties and studied the relationships between β*-open set and other topological sets including β*-continuity in topological spaces. We introduce and investigate several properties and characterizations of a new class of functions between topological spaces called β*- open, β*- closed, β*- continuous and β*- irresolute functions in topological spaces. We also introduce slightly β*- continuous, totally β*- continuous and almost β*- continuous functions between topological spaces and establish several characterizations of these new forms of functions. Furthermore, we also introduce and investigate certain ramifications of contra continuous and allied functions, namely, contra β*- continuous, and almost contra β*-continuous functions along with their several properties, characterizations and natural relationships. Moreover, we introduce new types of closed graphs by using β*- open sets and investigate its properties and characterizations in topological spaces.

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Fujita, Hiroshi, and Dimitri Shakhmatov. "Topological groups with dense compactly generated subgroups." Applied General Topology 3, no. 1 (2002): 85. http://dx.doi.org/10.4995/agt.2002.2115.

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<p>A topological group G is: (i) compactly generated if it contains a compact subset algebraically generating G, (ii) -compact if G is a union of countably many compact subsets, (iii) <sub>0</sub>-bounded if arbitrary neighborhood U of the identity element of G has countably many translates xU that cover G, and (iv) finitely generated modulo open sets if for every non-empty open subset U of G there exists a finite set F such that F U algebraically generates G. We prove that: (1) a topological group containing a dense compactly generated subgroup is both <sub>0</sub>-bounded and finitely generated modulo open sets, (2) an almost metrizable topological group has a dense compactly generated subgroup if and only if it is both <sub>0</sub>-bounded and finitely generated modulo open sets, and (3) an almost metrizable topological group is compactly generated if and only if it is -compact and finitely generated modulo open sets.</p>

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Selvaraj, Ganesan. "On micro forms of Delta-open sets, Delta-continuous maps and generalized micro Delta-continuous in micro topological spaces." Asia Mathematika 8, no. 2 (2024): 23——30. https://doi.org/10.5281/zenodo.13948738.

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This article introduces and studied m Delta-open sets in micro topological spaces. We offer a new class of sets called gm Delta-closed sets in micro topological spaces and we study some of its basic properties. The idea of introducing new classes of open sets are called micro semi Delta-open, micro alpha-Delta-open,  micro pre Delta-open,  micro b-Delta-open micro beta-Delta-open, micro regular Delta-open sets, micro semi Delta-continuous, micro alpha-Delta-continuous,  micro pre Delta-continuous, micro b-Delta-continuous micro beta-Delta-continuous and micro regular Delta-continuous in micro topological spaces.  We introduce m Delta-continuous maps, gm Delta-continuous maps, m Delta-irresolute maps, gm Delta-irresolute maps, contra m Delta-continuous  maps and contra gm Delta-continuous  maps in micro topological spaces and discuss some of their properties.   

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Al-omeri, Wadei Faris, Mohammed S. Md Noorani, Ahmad Al-Omari, and Takashi Noiri. "New Types of Multifunctions in ideal Topological Spaces via $e$-$\I$-Open sets and $\delta\beta$-$\I$-Open Sets." Boletim da Sociedade Paranaense de Matemática 34, no. 1 (2016): 213–23. http://dx.doi.org/10.5269/bspm.v34i1.26193.

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The purpose of the present paper is to introduce and investigate two new classes of continuous multifunctions called upper/lower $e$-$\I$-continuous multifunctions and upper/lower $\delta\beta_I$-continuous multifunctions by using the concepts of $e$-$\I$-open sets and $\delta\beta_I$-open sets. The class of upper/lower $e$-$\I$-continuous multifunctions is contained in that of upper/lower $\delta\beta_I$-continuous multifunctions. Several characterizations and fundamental properties concerning upper/lower $e$-$\I$-continuity and upper/lower $\delta\beta_I$-continuity are obtained.

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Sivaraj. "On \omega-Open and g\omega-Closed Sets." Journal of Advanced Research in Pure Mathematics 6, no. 2 (2014): 104–9. http://dx.doi.org/10.5373/jarpm.1839.090613.

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Chandrasekar, V., and G. Anandajothi. "$\theta g^{'''}$-Open sets in fuzzy topological spaces." Malaya Journal of Matematik S, no. 1 (2020): 524–29. http://dx.doi.org/10.26637/mjm0s20/0100.

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Balasubramaniyan, K., and R. Prabhakaran. "\(\widetilde{g}\)-Open Sets in Fuzzy Topological Spaces." Communications in Mathematics and Applications 14, no. 2 (2023): 721–26. http://dx.doi.org/10.26713/cma.v14i2.1760.

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Zhou, Huan, O. G. Hammad, and Ahmed Mostafa Khalil. "On Q p -Closed Sets in Topological Spaces." Journal of Mathematics 2022 (February 18, 2022): 1–10. http://dx.doi.org/10.1155/2022/9352861.

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In the present paper, we will propose the novel notions (e.g., Q p -closed set, Q p -open set, Q p -continuous mapping, Q p -open mapping, and Q p -closed mapping) in topological spaces. Then, we will discuss the basic properties of the above notions in detail. The category of all Q p -closed (resp. Q p -open) sets is strictly between the class of all preclosed (resp. preopen) sets and g p -closed (resp. g p -open) sets. Also, the category of all Q p -continuity (resp. Q p -open ( Q p -closed) mappings) is strictly among the class of all precontinuity (resp., preopen (preclosed) mappings) and g p -continuity (resp. g p -open ( g p -closed) mappings). Furthermore, we will present the notions of Q p -closure of a set and Q p -interior of a set and explain some of their fundamental basic properties. Several relations are equivalent between two different topological spaces. The novel two separation axioms (i.e., Q p - ℝ 0 and Q p - ℝ 1 ) based on the notion of Q p -open set and Q p -closure are investigated. The space of Q p - ℝ 0 (resp., Q p - ℝ 1 ) is strictly between the spaces of pre- ℝ 0 (resp., pre- ℝ 1 ) and g p - ℝ o (resp., g p - ℝ 1 ). Finally, some relations and properties of Q p - ℝ 0 and Q p - ℝ 1 spaces are explained.

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Abd El-latif, A. M. "Soft connectedness and irresoluteness via $$\beta $$ β -open soft sets". Afrika Matematika 28, № 5-6 (2017): 805–21. http://dx.doi.org/10.1007/s13370-017-0485-9.

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Aljarrah, H. H., and Mohd Salmi Md Noorani. "Separation Axioms Via \(\omega \beta\)-Open Sets in Bitopological Spaces." Journal of Advanced Studies in Topology 3, no. 3 (2012): 32. http://dx.doi.org/10.20454/jast.2012.305.

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