International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The aim of this paper is to introduce pgrw-closed maps and pgrw*-closed maps and to obtain some of their properties. In section 3 pgrw-closed map is defined and compared with other closed maps. In section 4 composition of pgrw-maps is studied. In section 5 pgrw*-closed maps are defined.
μ-πrα Closed Sets in Bigeneralized Topological SpacesIJERA Editor
The aim of the paper is to introduce the concept of μ(m,n)-πrα closed sets in bigeneralized topological spaces and study some of their properties. We also introduce the notion of μ(m,n)-πrα continuous function and μ(m,n)-πrα T1/2 spaces on bigeneralized topological spaces and investigate some of their properties. Mathematics subject classification: 54A05, 54A10
In this paper, we introduce the concepts of πgθ-closed map, πgθ-open map, πgθ-
homeomorphisms and πgθc-homeomorphisms and study their properties. Also, we discuss its relationship
with other types of functions.
Mathematics Subject Classification: 54E55
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
In this paper based on recently introduced approach we formulated some recommendations to optimize
manufacture drift bipolar transistor to decrease their dimensions and to decrease local overheats during
functioning. The approach based on manufacture a heterostructure, doping required parts of the heterostructure
by dopant diffusion or by ion implantation and optimization of annealing of dopant and/or radiation
defects. The optimization gives us possibility to increase homogeneity of distributions of concentrations
of dopants in emitter and collector and specific inhomogenous of concentration of dopant in base and at the
same time to increase sharpness of p-n-junctions, which have been manufactured framework the transistor.
We obtain dependences of optimal annealing time on several parameters. We also introduced an analytical
approach to model nonlinear physical processes (such as mass- and heat transport) in inhomogenous media
with time-varying parameters.
A polycycle is a 2-connected plane locally finite graph G with faces partitioned
in two faces F1 and F2. The faces in F1 are combinatorial i-gons.
The faces in F2 are called holes and are pair-wise disjoint.
All vertices have degree {2,...,q} with interior vertices of degree q.
Polycycles can be decomposed into elementary polycycles. For some parameters (i,q) the elementary polycycles can be classified and this allows to solve many different combinatorial problems.
In the present paper , we introduce and study the concept of gr- Ti- space (for i =0,1,2) and
obtain the characterization of gr –regular space , gr- normal space by using the notion of gr-open
sets. Further, some of their properties and results are discussed.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The aim of this paper is to introduce pgrw-closed maps and pgrw*-closed maps and to obtain some of their properties. In section 3 pgrw-closed map is defined and compared with other closed maps. In section 4 composition of pgrw-maps is studied. In section 5 pgrw*-closed maps are defined.
μ-πrα Closed Sets in Bigeneralized Topological SpacesIJERA Editor
The aim of the paper is to introduce the concept of μ(m,n)-πrα closed sets in bigeneralized topological spaces and study some of their properties. We also introduce the notion of μ(m,n)-πrα continuous function and μ(m,n)-πrα T1/2 spaces on bigeneralized topological spaces and investigate some of their properties. Mathematics subject classification: 54A05, 54A10
In this paper, we introduce the concepts of πgθ-closed map, πgθ-open map, πgθ-
homeomorphisms and πgθc-homeomorphisms and study their properties. Also, we discuss its relationship
with other types of functions.
Mathematics Subject Classification: 54E55
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
In this paper based on recently introduced approach we formulated some recommendations to optimize
manufacture drift bipolar transistor to decrease their dimensions and to decrease local overheats during
functioning. The approach based on manufacture a heterostructure, doping required parts of the heterostructure
by dopant diffusion or by ion implantation and optimization of annealing of dopant and/or radiation
defects. The optimization gives us possibility to increase homogeneity of distributions of concentrations
of dopants in emitter and collector and specific inhomogenous of concentration of dopant in base and at the
same time to increase sharpness of p-n-junctions, which have been manufactured framework the transistor.
We obtain dependences of optimal annealing time on several parameters. We also introduced an analytical
approach to model nonlinear physical processes (such as mass- and heat transport) in inhomogenous media
with time-varying parameters.
A polycycle is a 2-connected plane locally finite graph G with faces partitioned
in two faces F1 and F2. The faces in F1 are combinatorial i-gons.
The faces in F2 are called holes and are pair-wise disjoint.
All vertices have degree {2,...,q} with interior vertices of degree q.
Polycycles can be decomposed into elementary polycycles. For some parameters (i,q) the elementary polycycles can be classified and this allows to solve many different combinatorial problems.
In the present paper , we introduce and study the concept of gr- Ti- space (for i =0,1,2) and
obtain the characterization of gr –regular space , gr- normal space by using the notion of gr-open
sets. Further, some of their properties and results are discussed.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Los 12 expertos que contarán la historia del conflicto en La HabanaCrónicas del despojo
La Comisión Histórica del Conflicto, que se instaló hoy, tendrá la misión de establecer los orígenes y las causas del conflicto, los responsables y los impactos que ha tenido en la población civil.
El efecto de Ryanair sobre los destinos españolesHosteltur
La aerolínea low cost Ryanair es a día de hoy la compañía aérea que más reacciones a favor y en contra genera en España. Para unos, su modelo de expansión apoyado en subvenciones constituye una peligrosa apuesta por parte de los destinos aunque para otros Ryanair es sinónimo de crecimiento turístico. Algunos ven en esta aerolínea la encarnación del peor servicio posible, mientras que otros valoran sus bajas tarifas.
Indian Dental Academy: will be one of the most relevant and exciting training center with best faculty and flexible training programs for dental professionals who wish to advance in their dental practice,Offers certified courses in Dental implants,Orthodontics,Endodontics,Cosmetic Dentistry, Prosthetic Dentistry, Periodontics and General Dentistry.
In general topology many strong and weak forms of open and closed sets have been defined
and studied. Govindappa Navalagi introduced the concept of semi α-open sets which is a weaker form of
α-open sets. Semi*α-open set is defined analogously by replacing the closure operator by the
generalized closure operator due to Dunham in the definition of semi α-open sets. In this paper we
introduce a new class of sets, namely semi*α-closed sets, as the complement of semi*α-open sets. We
find characterizations of semi*α-closed sets. We also define the semi*α-closure of a subset. Further we
investigate fundamental properties of the semi*α-closure. We define the semi*α-derived set of a subset
and study its properties.
In this paper we introduce a new class of sets known as
δ
ˆ
S –closed sets in ideal topological
spaces and we studied some of its basic properties and characterizations. This new class of sets lies
between –I–closed [19] sets and g–closed sets, and its unique feature is it forms topology and it is
independent of open sets.
On Fuzzy - Semi Open Sets and Fuzzy - Semi Closed Sets in Fuzzy Topologic...IOSR Journals
Abstract: The aim of this paper is to introduce the concept of fuzzy - semi open and fuzzy - semi closed sets of a fuzzy topological space. Some characterizations are discussed, examples are given and properties are established. Also, we define fuzzy - semi interior and fuzzy - semi closure operators. And we introduce fuzzy
- t-set, -SO extremely disconnected space analyse the relations between them.
MSC 2010: 54A40, 03E72.
Continuous And Irresolute Functions Via Star Generalised Closed SetsIJMERJOURNAL
ABSTRACT: In this paper, we introduce a new class of continuous functions called semi*δ-continuous function and semi* δ-irresolute functions in topological spaces by utilizing semi* δ-open sets and to investigate their properties.
On Decomposition of gr* - closed set in Topological Spacesinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
RW-CLOSED MAPS AND RW-OPEN MAPS IN TOPOLOGICAL SPACESEditor IJCATR
In this paper we introduce rw-closed map from a topological space X to a topological space Y as the image
of every closed set is rw-closed and also we prove that the composition of two rw-closed maps need not be rw-closed
map. We also obtain some properties of rw-closed maps.
Notions via β*-open sets in topological spacesIOSR Journals
In this paper, first we define β*-open sets and β*-interior in topological spaces.J.Antony Rex Rodrigo[3] has studied the topological properties of 𝜂 * -derived, 𝜂 * -border, 𝜂 * -frontier and 𝜂 * exterior of a set using the concept of 𝜂 * -open following M.Caldas,S.Jafari and T.Noiri[5]. By the same technique the concept of β*-derived, β*-border, β*-frontier and β*exterior of a set using the concept of β*-open sets are introduced.Some interesting results that shows the relationships between these concepts are brought about
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
On Zα-Open Sets and Decompositions of ContinuityIJERA Editor
In this paper, we introduce and study the notion of Zα-open sets and some properties of this class of sets are investigated. Also, we introduce the class of A *L-sets via Zα-open sets. Further, by using these sets, a new decompositions of continuous functions are presented. (2000) AMS Subject Classifications: 54D10; 54C05; 54C08.
Similar to On (1,2)*-πgθ-CLOSED SETS IN BITOPOLOGICAL SPACES (20)
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
1. ISSN (e): 2250 – 3005 || Vol, 04 || Issue, 8 || August – 2014 ||
International Journal of Computational Engineering Research (IJCER)
www.ijceronline.com Open Access Journal Page 6
On (1,2)*-πgθ-CLOSED SETS IN BITOPOLOGICAL SPACES
C.Janaki1 , M. Anandhi2
1,2Asst. Professor, Dept. of Mathematics, L.R.G Government Arts College For Women,
Tirupur-4, India.
I. INTRODUCTION
Velicko[24] introduced the notions of θ-open subsets, θ-closed subsets and θ-closure, for the sake of
studying the important class of H-closed spaces in terms of arbitrary filterbases. Dontchev and Maki [7] alone
have explored the concept of θ-generalized closed sets. Regular open sets have been introduced and investigated
by Stone [23]. Levine [4,14] introduced generalized closed sets and studied their properties. Bhattacharya and
Lahiri [5], Arya and Nour [4], Maki et al.[15],[16] introduced semi-generalized closed sets, generalized semi-closed
sets and α–generalized closed sets and generalized α–closed sets respectively. O.Ravi et al [21] have
introduced the concepts of (1,2)*-semi-open sets, (1,2)*-α-open sets, (1,2)*-semi-generalized-closed sets and
(1,2)*-α-generalized closed sets in bitopological spaces. This paper is an attempt to highlight a new type of
generalized closed sets called (1,2)*-π generalized θ-closed (briefly (1,2)*-πgθ-closed) sets and a new class of
generalized functions called (1,2)*-πgθ-continuous functions and (1,2)*-πgθ-irresolute functions. These findings
result in procuring several characterizations of (1,2)*-πgθ-closed sets and as well as their application which
leads to an introduction of a new space called (1,2)*-πgθ-T½ space.
II. PRELIMINARIES
Throughout this paper (X,τ1,τ2)and (Y,σ1,σ2) represent bitopological spaces on which no separation
axioms are assumed unless otherwise mentioned.
Definition 2.1 ([20]). A subset S of a bitopological space (X,τ1,τ2) is said to be τ1,2-open if S = A B where
A τ1and B τ2. A subset S of X is τ1,2 - closed if the complement of S is τ1,2-open.
Definition 2.2 ([20]). Let S be a subset of X. Then
(i) The τ1τ2–interior of S , denoted by τ1τ2 –int (S) is defined by {G/G S
and G is τ1,2 -open } .
(ii) The τ1τ2–closure of S denoted by τ1τ2-cl(S) is defined by ∩{ F / S F and F is τ1,2 -closed}.
Definition 2.3 . A subset A of a bitopological space (X,τ1,τ2) is called
1. (1,2)* - semi-open[20] if A τ1τ2-cl(τ1τ2- int(A)).
2. (1,2)*- preopen [20] if A τ1τ2- int (τ1τ2-cl(A)). 3. (1,2)*-
α-open [20] if A τ1τ2- int (τ1τ2-cl (τ1τ2- int(A))).
4. (1,2)*-generalised closed (briefly (1,2)*-g-closed) [20] if τ1τ2-cl (A) U whenever A
U and U is τ1,2 -open in X.
5. (1,2)*-regular open[20] if A= τ1τ2- int (τ1τ2-cl(A)).
6. (1,2)* - semi-generalised-closed (briefly (1,2)*-sg-closed) [20] if (1,2)*-scl(A)
U whenever A U and U is (1,2)*-semi-open in X.
7. (1,2)*-generalized semi-closed (briefly (1,2)*-gs-closed)[20] if (1,2)*-scl(A) U,
whenever A U and U is τ1,2-open in X .
Abstract
In this paper we introduce a new class of sets called (1,2)*-πgθ-closed sets in bitopological spaces.
Also we find some basic properties of (1,2)*-πgθ-closed sets. Further, we introduce a new space called
(1,2)*-πgθ-T½ space. Mathematics Subject Classification: 54E55, 54C55
KEY WORDS: (1,2)*-πgθ-closed set, (1,2)*-πgθ-open set, (1,2)*-πgθ-continuity and (1,2)*-πgθ-T½ space.
2. On (1,2)*-Πgθ-Closed Sets…
www.ijceronline.com Open Access Journal Page 7
8. (1,2)*-α-generalized-closed (briefly (1,2)*-αg-closed) [20] if (1,2)*-αcl(A) U,
whenever A U and U is τ1,2 -open in X.
9. (1,2)*-generalized α-closed (briefly (1,2)*-gα-closed)[20] if (1,2)*- αcl(A) U,
whenever A U and U is (1,2)*-α -open in X.
10. a (1,2)*-θ-generalized closed (briefly, (1,2)*-θg-closed) set [11] if (1,2)*-clθ(A) U
whenever A U and U is τ1,2 -open in (X,τ1,τ2).
11. (1,2)*-π generalized closed(briefly (1,2)*-πg-closed [8] if (1,2)*-cl(A) U, whenever A U
and U is τ1,2 -π-open.
12. (1,2)*-π generalized α-closed (briefly (1,2)*-πgα-closed)[3] if (1,2)*-αcl(A) U, whenever
A U and U is τ1,2 -π-open.
13. (1,2)*-π generalized semi-closed (briefly (1,2)*-πgs-closed)[4] if (1,2)*-scl(A) U,
whenever A U and U is τ1,2 -π-open.
14. (1,2)*-π generalized b-closed (briefly (1,2)*-πgb-closed)[22] if (1,2)*-bcl(A) U, whenever
A U and U is τ1,2 -π-open.
15. (1,2)*-π generalized pre-closed( briefly (1,2)*-πgp-closed)[19] if (1,2)*-pcl(A) U,
whenever A U and U is τ1,2 -π-open.
The complement of a (1,2)*-semi-closed (resp. (1,2)*-α-closed, (1,2)*-g-closed, (1,2)*-sg-closed, (1,2)*-
gs-closed, (1,2)*-αg-closed (1,2)*-gα-closed, (1,2)*-θg-closed , (1,2)* πg-closed, (1,2)*- πgα-closed, (1,2)*-
πgs-closed, (1,2)*-πgb-closed, (1,2)*-πgp-closed) set is called (1,2)*-semi open(resp. (1,2)*-α-open,(1,2)*-g-open,(
1,2)*-sg-open,(1,2)*-gs-open,(1,2)*-αg-open, (1,2)*-gα-open, (1,2)*-θg-closed, (1,2)*-πg-open,(1,2)*-
πgα-open,(1,2)*-πgs-open,(1,2)*-πgb-open, (1,2)*-πgp-open).
Definition 2.4 The finite union of (1,2)*-regular open sets[5] is said to be τ1,2-π-open. The
complement of τ1,2-π-open is said to be τ1,2-π-closed.
Definition2. 5: A function f: (X,τ1,τ2) →(Y,σ1,σ2) is called
(i) τ1,2-π-open map[3] if f(F) is τ1τ2-π-open map in Y for every τ1,2-openset F in X.
(ii) (1,2)*-θ-continuous[7] if f-1 (V) is (1,2)*-θ-closed in (X,τ1,τ2) for every (1,2)*-closed
set V in (Y,σ1,σ2).
(iii) (1,2)*-θ-irresolute[7] if f-1 (V) is (1,2)*-θ-closed in (X,τ1,τ2) for every (1,2)*-θ-closed
set V in (Y,σ1,σ2).
III. (1,2)*- πgθ-closed set
We introduce the following definition.
Definition 3.1. A subset A of (X,τ1,τ2) is called(1,2)*- π generalized θ-closed set (briefly (1,2)*-πgθ-closed ) if
τ1τ2-clθ(A) U whenever A U and U is τ1τ2 -π-open.
The complement of (1,2)*-πgθ-closed is (1,2)*-πgθ-open..
Theorem 3.2:
1. Every (1,2)*-θ- closed set is (1,2)*-πgθ-closed.
2. Every (1,2)*-θg-closed set is (1,2)*-πgθ-closed.
3. Every (1,2)*-πgθ-closed set is (1,2)*-πg-closed.
4. Every (1,2)*-πgθ-closed set is (1,2)*-πgα -closed.
5. Every (1,2)*-πgθ-closed set is (1,2)*-πgs -closed.
6. Every (1,2)*-πgθ-closed set is (1,2)*-πgb-closed.
7. Every (1,2)*-πgθ-closed set is (1,2)*-πgp-closed
Proof: Straight forward.
Converse of the above need not be true as seen in the following examples.
Example 3.3 Let X ={a,b,c}.τ1={ ϕ, X ,{a},{c},{a,c} }; τ2={ϕ, X,{ b,c }}:
Let A ={b}. Then A is πgθ-closed but not θ-closed.
Example3.4 Let X={a,b,c,d}.τ1={ϕ,{b},{d},{b,d},{a,b,d},X}; τ2={ϕ,{b,d},{b,c,d}, X}. Let A = {a,d}.
Then A is (1,2)*-πgθ-closed but not (1,2)*-θg-closed.
3. On (1,2)*-Πgθ-Closed Sets…
www.ijceronline.com Open Access Journal Page 8
Example 3.5 Let X = { a,b,c,d}, τ1 = {ϕ,{a},{b},{a,b},{a,b,d},X }, τ2={ ϕ ,{a,c}, {a,b,c},X}. Let A = {c}. Then A is (1,2)*-πg-closed but not (1,2)*-πgθ-closed. Example 3.6 Let X = {a,b,c,d}. τ1= {ϕ,{a},{d},{a,d}, X }; τ2 ={ ϕ,{c,d}, {a,c,d}, X}: Let A ={c}. Then A is (1,2)*-πgα-closed set but not (1,2)*-πgθ-closed. Example3.7 Let X = {a,b,c,d}. τ1= {ϕ,{a},{b},{a,b}, X}; τ2 = { ϕ,{a,b}, {a,b,d}, X,}; Let A ={a}. Then A is (1,2)*-πgs-closed set but not (1,2)*-πgθ-closed. Example 3.8 Let X={a,b,c,d}.τ1= {ϕ,{a},{d},{a,d}, X }; τ2 = {ϕ,{c,d}, {a,c,d}, X}; Let A ={a,c}. Then A is (1,2)*-πgb-closed set but not (1,2)*-πgθ-closed. Example 3.9 Let X = {a,b,c,d}. τ1= {ϕ,{a},{d},{a,d}, X }; τ2 ={ ϕ,{c,d},{a,c,d}, X}: Let A = {c}. Then A is (1,2)*-πgp-closed but not (1,2)*-πgθ-closed. Remark 3.10 The above discussions are summarized in the following diagram.
. Remark 3.11 (1,2)*-πgθ-closed is independent of (1,2)*-closedness, (1,2)*-α-closedness, (1,2)*-semi- closedness, (1,2)*-sg-closedness, (1,2)*-gs-closedness, (1,2)*-g-closedness, (1,2)*-αg-closedness and(1,2)*-gα- closedness, as seen in the following examples. Example 3.12 Let X = {a,b,c,d}, τ1= {ϕ,{a},{b},{a,b}, X }; τ2 = {ϕ, {a,b,d}, X}: Let A = {d}. Then A is (1,2)*-πgθ-closed but not (1,2)*-g-closed. Example 3.13 Let X = {a,b,c,d,e}, τ1= {ϕ,{a,b},{a,b,c,d}, X}; τ2= {ϕ,{c,d},X}: Let A = {e}. Then A is (1,2)*-g-closed but not (1,2)*-πgθ-closed. Example3.14. Let X = {a,b,c,}, τ1= { ϕ,{a},{b},{a,b}, X }; τ2 = {ϕ, {b,c}, X}: Let A ={b}. Then A is (1,2)*- πgθ-closed but not (1,2)*-closed, (1,2)*-α-closed, (1,2)*-semi-closed. Example 3.15 Let X = {a,b,c}, τ1= {ϕ,{a},{b},{a,b}, X }; τ2 = { ϕ, {b,c}, X}: Let A ={a}. Then A is (1,2)*- closed, (1,2)*-α-closed, (1,2)*-semi-closed but not (1,2)*-πgθ-closed. Example 3.16 Let X = {a,b,c,d}, τ1= {ϕ,{a},{b},{a,b},{a,b,c}, X }; τ2 = {ϕ, {a,b,d}, X}: (i) Let A={a,b,c}. Then A is (1,2)*-πgθ-closed but neither (1,2)*-sg-closed nor (1,2)*-gs-closed. (ii) Let A = {a}. Then A is both (1,2)*-sg-closed and (1,2)*-gs-closed but not (1,2)*-πgθ-closed. Example 3.17 Let X = {a,b,c,d}, τ1={ϕ,{c,d},{a,c,d}, X), τ2={ϕ,{a},{d},{a,d},{c,d},X}. (i) Let A={b,d}. Then A is (1,2)*-πgθ-closed but neither (1,2)*-αg-closed nor (1,2)*-gα- closed. (ii)Let A ={c}.Then A is (1,2)*-αg-closed,(1,2)*-gα-closed but not (1,2)*-πgθ-closed. Remark 3.18 The above discussions are summarized in the following diagram.
(1,2)*-πgθ-closed
(1,2)*-θ-closed
(1,2)*-θg-closed
(1,2)*-πg-closed
(1,2)*-πgα-closed
(1,2)*-πgb-closed
(1,2)*-πgp-closed
(1,2)*-πgs-closed
4. On (1,2)*-Πgθ-Closed Sets…
www.ijceronline.com Open Access Journal Page 9
Remark 3.19 A finite union of (1,2)*-πgθ -closed sets is always a (1,2)*-πgθ-closed.
Proof: Let A, B ϵ (1,2)*-πGθC(X). Let U be any τ1τ2-π-open set such that (A B) U. Since (1,2)*-
clθ(A B) = (1,2)*-clθ(A) (1,2)*-clθ(B) U U=U. This implies (1,2)*-clθ(A B) U. Hence A B is
also a (1,2)*-πgθ-closed set.
Remark 3.20 The intersection of two (1,2)*-πgθ-closed sets need not be (1,2)*-πgθ-closed as seen in the
following example.
Example3.21 Let X = {a,b,c,d}, τ1 = {ϕ,{a},{b},{a,b},X }; τ2 = {ϕ,{a,b,d},X}:
Let A={a,b,c} and B ={ a,b,d}.Clearly A and B are (1,2}*-πgθ-closed sets. But A∩B = { a,b} is not a (1,2)*-
πgθ-closed set.
Proposition 3.22 Let A be (1,2)*-πgθ-closed in (X,τ). Then (1,2)*-clθ(A)–A does not contain any non-empty
τ1τ2-π-closed set.
Proof: Let F be a non-empty τ1τ2-π-closed set such that F (1,2)*-clθ(A)–A. Since A is (1,2)*-πgθ-closed,
A X–F where X–F is τ1τ2-π-open implies (1,2)*-clθ(A) (X–F). Hence F X ̶ (1,2)*-clθ(A). Now,
F (1,2)*-clθ(A)∩X–(1,2)*-clθ(A) implies F= ϕ which is a contradiction. Therefore clθ(A)–A does not contain
any non-empty τ1τ2-π-closed set.
Remark 3.23 The converse of Proposition 3.22 need not be true as shown in the following example.
Example 3.24 Let X = {a,b,c}. τ1= {ϕ, X,{b} }; τ2 = { ϕ, X,{c}}:
Let A = {b,c}. Then (1,2)*-clθ(A) ̶ A contains no non-empty τ1τ2-π-closed set. However A is not (1,2)*-πgθ-
closed.
Proposition 3.25 If A is a τ1,2-regular open and (1,2)*-πgθ-closed subset of (X,τ1,τ2), then A is a (1,2)*-θ-
closed subset of (X,τ1,τ2).
Proof. Since A is τ1,2-regular open and (1,2)*-πgθ-closed, (1,2)*-clθ(A) A . Hence A is (1,2)*-θ-closed.
Proposition3.26 Let A be a (1,2)*-πgθ-closed subset of (X,τ1, τ2). If A B (1,2)*- clθ(A), then B is also a
(1,2)*-πgθ-closed subset of (X,τ1,τ2).
Proof. Let U be a τ1τ2-π-open set of (X,τ1, τ2) such that B U. Then A U. Since A is a (1,2)*-πgθ-closed
set, (1,2)*-clθ(A) U. Also since B (1,2)*-clθ(A), (1,2)*-clθ(B) (1,2)* - clθ((1,2)*-clθ(A))=(1,2)*-
clθ(A).Thus (1,2)*-clθ(B) U. Hence B is also a (1,2)*-πgθ-closed subset of (X,τ1,τ2).
Theorem 3.27 Let A be a (1,2)*-πgθ-closed sub set in X. Then A is (1,2)*-θ-closed if and only if (1,2)*-
clθ(A)–A is τ1,2 -π-closed.
Proof. Necessity: Let A be (1,2)*-θ-closed subset of X. Then (1,2)*-clθ(A)=A and (1,2)*-clθ(A)–A=ϕ which
is τ1,2 -π-closed.
Sufficiency: Since A is (1,2)*-πgθ-closed, by proposition 3.22 (1,2)*-clθ(A–A contains no non-empty τ1,2- π-
closed set. But (1,2)*-clθ(A) – A is τ1,2-π-closed. This implies (1,2)*-clθ(A) – A=ϕ, which means (1,2)*-
clθ(A)=A and hence A is (1,2)*-θ-closed.
5. On (1,2)*-Πgθ-Closed Sets…
www.ijceronline.com Open Access Journal Page 10
IV. (1,2)*-πgθ -open sets
Definition 4.1 A subset A of (X, τ1, τ2) is called (1,2)*-πgθ-open if and only if Ac is (1,2)*-πgθ-closed in
(X,τ1,τ2).
Remark 4.2 For a subset A of (X, τ1, τ2), (1,2)*-clθ(Ac) = [(1,2)*- intθ(A)]c
Theorem 4.3 A subset A of (X,τ1,τ2) is (1,2)*-πgθ-open if and only if F (1,2)*-intθ(A) whenever F is τ1 τ2-π-
closed and F A.
Proof. Necessity: Let A be a (1,2)*-πgθ-open set in (X,τ1,τ2). Let F be τ1τ2-π-closed and F A. Then Fc
Ac
and Fc is τ1τ2-π-open. Since Ac is (1,2)*-πgθ-closed, (1,2)*-clθ(Ac) Fc. By remark 4.2, [(1,2)*-intθ(A)]c
Fc.
That is F (1,2)*-intθ(A).
Sufficiency: Let Ac
U where U is τ1τ2-π-open. Then Uc A where Uc is (1,2)*-τ1τ2-π-closed. By hypothesis
Uc (1,2)*-intθ(A ). That is [(1,2)*-intθ(A}]c
U. By remark 4.2, (1,2)*-clθ(Ac) U. This implies Ac is
(1,2)*-πgθ-closed. Hence A is (1,2)*-πgθ-open.
Theorem 4.4 If (1,2)*-intθ(A) B A and A is (1,2)*-πgθ-open, then B is also (1,2)*-πgθ-open.
Proof. (1,2)*-intθ(A) B A implies Ac
Bc
[(1,2)*-intθ(A)]c . By remark 4.2 Ac
Bc
(1,2)*-clθ(Ac).
Also Ac is (1,2)*-πgθ-closed. By Proposition 3.26 Bc is (1,2)*-πgθ-closed. Hence B is (1,2)*-πgθ-open.
As an application of (1,2)*-πgθ-closed sets we introduce the following definition.
Definition 4.5 A space (X,τ1, τ2) is called a (1,2)*-πgθ-T½ space if every (1,2)*-πgθ-closed set is (1,2)*-θ-
closed.
Theorem 4.6 For a space (X,τ1,τ2) the following conditions are equivalent.
(i) (X,τ1,τ2) is a (1,2)*-πgθ-T½ space.
(ii) Every singleton set of (X,τ1,τ2) is either τ1τ2-π-closed or (1,2)*-θ-open.
Proof. (i) (ii). Let x X. Suppose {x} is not a τ1τ2-π- closed set of (X, τ1, τ2). Then X–{x} is not a τ1τ2-π-
open set. So X is the only τ1τ2-π-open set containing X–{x}.So X-x is a (1,2)*-πgθ-closed set of (X,τ1,τ2). Since
(X,τ1,τ2) is a (1,2)*-πgθ-T½ space, X– x is a (1,2)*-θ-closed set of (X,τ1,τ2) or equivalently {x} is a (1,2)*-θ-
open set of (X,τ1, τ2).
(ii) (i). Let A be a (1,2)*-πgθ-closed set of X. Trivially A (1,2)*-clθ(A). Let xϵ1,2)*-clθ(A). By (ii) {x} is
either τ1τ2-π-closed or (1,2)*-θ-open.
(a) Suppose that {x} is τ1 τ2-π-closed. If x A, then xє(1,2)*-clθ(A)–A contains a non-empty τ1τ2-π-closed set
{x}. By proposition 3.6 we arrive at a contradiction. Thus xϵA.
(b) Suppose that {x} is a (1,2)*-θ-open. Since xϵ(1,2)*-clθ(A)=A or equivalently A is (1,2)*-θ-closed. Hence
(X,τ1,τ2) is a (1,2)*-πgθ-T½ space.
V. (1,2)*-πgθ-continuous and (1,2)*-πgθ-irresolute functions
Definition 5.1 A function f:(X,τ1,τ2) →(Y,σ1,σ2) is called (1,2)*-πgθ-continuous if every f-1(V) is (1,2)*-πgθ-
closed in (X,τ1, τ2) for every(1,2)*-σ1σ2-closed set V of (Y,σ1,σ2).
Definition 5.2 A function f:(X,τ1,τ2)→(Y,σ1,σ2) is called (1,2)*-πgθ-irresolute if f-1 (V) is (1,2)*-πgθ-closed in
(X,τ1,τ2) for every (1,2)*-πgθ-closed set V in (Y,σ1,σ2).
Remark 5.3 (1,2)*-πgθ-irresolute function is independent of (1,2)*-θ-irresoluteness, as seen in the following
examples.
Example 5.4 Let X=Y={a,b,c,d}, τ1={ ϕ,{a},{d},{a,d},{a,c,d},X},τ2= {ϕ,{a,d},{a,b,d}, X}, σ1 =
{ϕ,{a},{d},{a,d},X},σ2= {ϕ,{c,d},{a,c,d},X}. Let f: (X,τ1,τ2)→(Y,σ1,σ2) be an identity function. Then f is
(1,2)*-θ-irresolute but not (1,2)*-πgθ-irresolute, since f-1 [{b,c,d}]={b,c,d} is not (1,2)*-πgθ-closed in (X,τ1,τ2).
6. On (1,2)*-Πgθ-Closed Sets…
www.ijceronline.com Open Access Journal Page 11
Example 5.5 Let X=Y={a,b,c,d}, τ1={ϕ,{a},{d},{a,d},{a,c,d},X }, τ2 = {ϕ,{a,d}, {a,b,d},X}, σ1
={ϕ,{a},{c},{a,c}, {c,d},{a,c,d},X},σ2= {ϕ,{d},{a,b,d},X}, Let f: (X,τ1,τ2)→(Y,σ1,σ2) be an identity function.
Then f is (1,2)*-πgθ-irresolute but not (1,2)*-θ-irresolute, since f-1 [{a,b,d}] = {a,b,d} is not (1,2)*-θ-closed in
(X,τ1,τ2).
Remark 5.6: Every (1,2)*-θ-continuous is (1,2)*-πgθ-continuous.
The converse of the above need not be true as seen in the following examples.
Example 5.7: Let X=Y={a,b,c,d,e }, τ1={ ϕ,{a,b},{a,b,c},X }, τ2= {ϕ,{c},X},σ1 ={ ϕ, {a,b}, {a,b,c,d}, X },σ2={
ϕ,{c,d},{a,b,c,d}, X }.Let f: (X,τ1,τ2)→(Y,σ1,σ2) be an identity function. Then f is (1,2)*-πgθ-continuous but
not (1,2)*-θ-continuous, since f-1[{c,d,e}]={c,d,e} is not (1,2)*-θ-closed in (X, τ1, τ2).
Remark 5.8 (1,2)*-πgθ-continuous is independent of (1,2)*- πgθ-irresolute as seen in the following examples.
Example 5.9 Let X=Y={a,b,c,d,e }, τ1={ ϕ,{a,b},{a,b,c,d},X }, τ2= {ϕ,{c,d},X},σ1 ={ ϕ,{b}, {b,c}, X },σ2={
ϕ,{c},{a,b},{a,b,c}, X }.Let f: (X,τ1,τ2)→(Y,σ1,σ2) be an identity function. Then f is (1,2)*-πgθ-continuous but
not (1,2)*-πgθ-irresolute, since f-1[{d}]={d} is not (1,2)*-πgθ-closed in (X, τ1, τ2) where {d}is πgθ-closed in
(Y,σ1,σ2).
Example 5.10: Let X=Y={a,b,c,d}, τ1={ϕ,{a},{b},{a,b},{a,b,c},X }, τ2= {ϕ,{b},{c},{b,c},{a,c},
{a,b,c},{a,b,d},X},σ1 ={ ϕ, {a},{b},{a,b},{a,d}, {a,b,d},X },σ2={ ϕ,{c},{a,c},{b,c},{a,b,c}, {a,c,d}, X }.Let f:
(X,τ1,τ2)→(Y,σ1,σ2) be an identity function. Then f is (1,2)*-πgθ-irresolute but not (1,2)*-πgθ-continuous, since
f-1[{b}]={b} is not (1,2)*-πgθ-closed in (X, τ1, τ2) where {b}is closed in (Y,σ1,σ2).
Remark 5.11 The above discussions are summarized in the following diagram.
Where (1) (1,2)*-θ-continuous; (2) (1,2)*-πgθ-continuous;
(3) (1,2)*-πgθ-irresolute; (4) (1,2)*-θ-irresolute.
Remark 5.12 Composition of two (1,2)*-πgθ-continuous function need not be (1,2)*-πgθ-continuous.
Example 5.13 Let X=Y=Z={a,b,c,d,e},τ1={ϕ,{a,b},{a,b,c},X}, τ2= {ϕ,{c},X},σ1 ={ϕ,{a,b},X}, σ2 ={
ϕ,{c,d},{a,b,c,d},X}, η1={ϕ,{a,b,c,d},X},η2 ={ ϕ ,{e}, X}. Let f:(X,τ1,τ2)→(Y,σ1,σ2) and g:(Y,σ1,σ2)→(Z,η1,η2)
be the identity functions. Both f and g are (1,2)*-πgθ-continuous but gof is not (1,2)*-πgθ-continuous, since
(gof)-1[{a,b,c,d}]={a,b,c,d} is not (1,2)*-πgθ-closed in (X,τ1,τ2).
Theorem 5.14 Let f: (X,τ1, τ2)→(Y,σ1,σ2) be a function .
(i) If f is (1,2)*-πgθ-irresolute and X is(1,2)*-πgθ-T½ space, then f is(1,2)*-θ-irresolute.
(ii) If f is (1,2)*-πgθ-continuous and X is (1,2)*-πgθ-T½ space then f is (1,2)*-θ-continuous.
Proof: (i) Let V be (1,2)*-θ-closed in Y. Since f is (1,2)*-πgθ-irresolute, f-1(V) is (1,2)*- πgθ-closed in X. Since
X is (1,2)*-πgθ-T½ space, f-1(V) is (1,2)*-θ-closed in X. Hence f is (1,2)*-θ-irresolute.
(ii)Let V be closed in Y. Since f is (1,2)*-πgθ-continuous, f-1(V) is (1,2)*-πgθ-closed in X. By
assumption, it is (1,2)*-θ-closed. Therefore f is (1,2)*-θ-continuous.
Theorem 5.15 If the bijective f: (X,τ1,τ2)→( Y,σ1,σ2) is (1,2)*-θ-irresolute and τ1τ2-π-open map, then f is
(1,2)*-πgθ-irresolute.
7. On (1,2)*-Πgθ-Closed Sets…
www.ijceronline.com Open Access Journal Page 12
Proof: Let V be (1,2)*-πgθ-closed in Y. Let f-1(V) U where U is τ1τ2-π-open in X. Then V f(U) and f(U) is
τ1τ2-π-open implies (1,2)*-clθ(V) f(U). This implies f-1( (1,2)*-clθ(V)) U. Since f is (1,2)*-θ-irresolute, f-
1((1,2)*-clθ(V)) is (1,2)*-θ-closed. Hence (1,2)*- clθ(f-1(V)) (1,2)*- clθ(f-1((1,2)*-clθ(V)))=f-1((1,2)*-
clθ(V)) U. Therefore f is (1,2)*-πgθ-irresolute.
Theorem 5.16 :If f: (X,τ1,τ2)→(Y,σ1,σ2) is (1,2)*-πgθ-irresolute map and g: (Y,σ1,σ2)→ (Z,η1,η2) is (1,2)*-
πgθ-continuous map, the composition gof: (X,τ1,τ2)→ (Z,η1,η2) is a (1,2)*-πgθ-continuous map.
Proof: Let V be η1η2–closed set in Z. Since g:(Y,σ1,σ2)→(Z,η1,η2) is a (1,2)*-πgθ-continuous map,g-1(V) is
(1,2)*-πgθ-closed in Y. By hypothesis ,f-1(g-1(V))=(gof)-1(V) is (1,2)*-πgθ-closed in X. Hence gof: (X,τ1,τ2) →
(Z,η1,η2) is a (1,2)*-πgθ-continuous map.
V. CONCLUSION
Through the above findings, this paper has attempted to compare (1,2)*-πgθ-closed with the other
closed sets in bitopological spaces. An attempt of this paper is to state that the several definitions and results that
shown in this paper, will result in obtaining several characterizations and enable to study various properties as
well. It brings to limelight that the weaker form of continuity in bitopological settings is the future scope of
study.
REFERENCES
[1] M. E. Abd El-Monsef, S.Rose Mary and M.Lellis Thivagar (1,2)*- α g
-closed sets in topological spaces, Assiut Univ. J. of
Mathematics and Computer Science, Vol.36(1) (2007), 43-51.
[2] M. Anandhi and C.Janaki: On πgθ-Closed sets in Topological spaces, International Journal of Mathematical Archive-3(11),
2012, 3941-3946.
[3] I. Arockiarani and K. Mohana, (1,2)*- πgα-closed sets and (1,2)*-Quasi-α-normal Spaces in Bitopological settings, Antarctica
j. Math.,7(3)(2010), 3465-355.
[4] S. P. Arya and T. Nour: Characterizations of S-normal spaces, Indian J. Pure. Appl. Math., 21(1990), No. 8, 717-719.
[5] P.Bhattacharya and B.K. Lahiri: Semi-generalised closed sets in topology, Indian J. Math., 29(1987), 375-382.
[7] J. Dontchev and H. Maki,On θ-generalized closed sets,Internat.J. Math& Math.Sci. 22(2) (1999) 239-249.
[8] J. Dontchev and T. Noiri, Quasi Normal Spaces and πg-closed sets, Acta Math. Hungar., 89(3)(2000). 211-219.
[9] J. Dontchev and M. Przemski, The various decompositions of continuous and weakly continuous functions, Acta Math.
Hungar., 71(1996), no. 1-2, 109-120.
[10] E. Ekici and M . Caldas, Slightly-continuous functions, Bol. Soc. Parana. Mat. (3) 22 (2004), 63-74.
[11] Govindappa Navalagi and Md. Hanif Page, θ-generalized semi-open and θ- generalized semi-closed functions Proyeceiones J.
of Math. 28,2 ,(2009) 111-123.
[12] C.Janaki, Studies on πgα-closed sets in Topology, Ph.D Thesis, Bharathiar University, Coimbatore (2009).
[13] N.Levine: Semi-open sets and semi-continuity in topological spaces, Amer . Math. Monthly, 70 (1963), 36-41.
[14] N.Levine: Generalised closed sets in topology, Rend. Circ. Mat. Palermo, 19(1970), 89-96.
[15] H. Maki,R.Devi and K.Balachandran: Semi-generalised closed maps and generalized semi-closed maps, Mem. Fac. Sci. Kochi
Univ. Ser. A Math.,14(1993),41-54.
[16] H. Maki, R. Devi and K. Balachandran generalized α-closed maps and α-generalised closed maps, Indian J. Pure . Appl. Math.,
29(1998), No. 1, 37-49.
[17] A.S. Mashhour, M. E. Abd El- Monsef and S.N. El Deeb: On pre continuous and weakly pre continuous mappings, Proc. Math.
Phys. Soc. Egypt., 53(1982), 47-53.
[18] O.Njastad: On some Classes of nearly open sets, Pacific J. Math.,15(1965),961-970.
[19] J.H. Park, On πgp-closed sets in topological spaces, Indian J. Pure. Appl. Math., (To appear).
[20] O.Ravi and M.Lellis Thivagar, On Stronger forms of (1,2)* - quotient mappings on bitopological spaces,Internat. J.Math.Game
Theory & Algebra4(2004),No.6,481-492.
[21] O.Ravi, M.Lellis Thivagarand M.E.Abd El-Monsef, “Remarks on bitopological (1,2)*-quotient mappings”,J. Egypt Math. Soc.
Vol. 16, No.1, pp.17-25,2008.
[22] D.Sreeja and C.Janaki, On(1,2)*-πgb-closed sets, Inter. J. of Computer Applicationz (0975-8887), Vol.42.,No.5.
[23] M.H. Stone : Application of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41 (1937), 375-381.
[24] N. V. Velicko, On H- closed topological spaces, Amer. Math. Soc. Transl.,78, (1968) 103-118.