This document introduces the concept of γ-sαg*-semi open sets in topological spaces and some of their properties. It begins by discussing previous related concepts like γ-open sets, γ-closure, and γ-semi open sets. It then defines what a γ-sαg*-semi open set is and establishes some basic properties. The main part of the document introduces and defines the concepts of γ-sαg*-semi Ti spaces for i=0, 1/2, 1, 2. It establishes properties of γ-sαg*-semi g-closed sets and proves several theorems about γ-sαg*-semi closure operators and their relationships to other concepts. The document contributes to the mathematical
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
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological SpacesIOSR Journals
In this paper we introduce and study the concept of a new class of closed sets called (𝜏𝑖, 𝜏𝑗)− regular generalized b- closed sets (briefly(𝜏𝑖, 𝜏𝑗)− rgb-closed) in bitopological spaces.Further we define and study new neighborhood namely (𝜏𝑖, 𝜏𝑗)− rgb- neighbourhood (briefly(𝜏𝑖, 𝜏𝑗)− rgb-nhd) and discuss some of their properties in bitopological spaces. Also, we give some characterizations and applications of it.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
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
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological SpacesIOSR Journals
In this paper we introduce and study the concept of a new class of closed sets called (𝜏𝑖, 𝜏𝑗)− regular generalized b- closed sets (briefly(𝜏𝑖, 𝜏𝑗)− rgb-closed) in bitopological spaces.Further we define and study new neighborhood namely (𝜏𝑖, 𝜏𝑗)− rgb- neighbourhood (briefly(𝜏𝑖, 𝜏𝑗)− rgb-nhd) and discuss some of their properties in bitopological spaces. Also, we give some characterizations and applications of it.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
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.
Abstract: In this paper, we define and study about a new type of generalized closed set called, g∗s-closed set.Its relationship with already defined generalized closed sets are also studied
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.
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
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.
Notes on intersection theory written for a seminar in Bonn in 2010.
Following Fulton's book the following topics are covered:
- Motivation of intersection theory
- Cones and Segre Classes
- Chern Classes
- Gauss-Bonet Formula
- Segre classes under birational morphisms
- Flat pull back
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
In this paper, we define and study about a new type of generalized closed set called, 𝑔∗s-closed set.Its relationship with already defined generalized closed sets are also studied.
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
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.
Abstract: In this paper, we define and study about a new type of generalized closed set called, g∗s-closed set.Its relationship with already defined generalized closed sets are also studied
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.
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
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.
Notes on intersection theory written for a seminar in Bonn in 2010.
Following Fulton's book the following topics are covered:
- Motivation of intersection theory
- Cones and Segre Classes
- Chern Classes
- Gauss-Bonet Formula
- Segre classes under birational morphisms
- Flat pull back
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
In this paper, we define and study about a new type of generalized closed set called, 𝑔∗s-closed set.Its relationship with already defined generalized closed sets are also studied.
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
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.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
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.
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.
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
On some locally closed sets and spaces in Ideal Topological SpacesIJMER
In this paper we introduce and characterize some new generalized locally closed sets
known as
δ
ˆ
s-locally closed sets and spaces are known as
δ
ˆ
s-normal space and
δ
ˆ
s-connected space and
discussed some of their properties
On (1,2)*-πgθ-CLOSED SETS IN BITOPOLOGICAL SPACESijceronline
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.
The determination of this paper is to introduce two new spaces , namely 𝑆𝑔
∗
-compact and 𝑆𝑔
∗
-
connected spaces. Additionally some properties of these spaces are investigated.
Mathematics Subject Classification: 54A05
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.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Search and Society: Reimagining Information Access for Radical FuturesBhaskar Mitra
The field of Information retrieval (IR) is currently undergoing a transformative shift, at least partly due to the emerging applications of generative AI to information access. In this talk, we will deliberate on the sociotechnical implications of generative AI for information access. We will argue that there is both a critical necessity and an exciting opportunity for the IR community to re-center our research agendas on societal needs while dismantling the artificial separation between the work on fairness, accountability, transparency, and ethics in IR and the rest of IR research. Instead of adopting a reactionary strategy of trying to mitigate potential social harms from emerging technologies, the community should aim to proactively set the research agenda for the kinds of systems we should build inspired by diverse explicitly stated sociotechnical imaginaries. The sociotechnical imaginaries that underpin the design and development of information access technologies needs to be explicitly articulated, and we need to develop theories of change in context of these diverse perspectives. Our guiding future imaginaries must be informed by other academic fields, such as democratic theory and critical theory, and should be co-developed with social science scholars, legal scholars, civil rights and social justice activists, and artists, among others.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
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.
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
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.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
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.
"Impact of front-end architecture on development cost", Viktor TurskyiFwdays
I have heard many times that architecture is not important for the front-end. Also, many times I have seen how developers implement features on the front-end just following the standard rules for a framework and think that this is enough to successfully launch the project, and then the project fails. How to prevent this and what approach to choose? I have launched dozens of complex projects and during the talk we will analyze which approaches have worked for me and which have not.
From Daily Decisions to Bottom Line: Connecting Product Work to Revenue by VP...
Gamma sag semi ti spaces in topological spaces
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.3, 2012
α
γ - sαg*-Semi Ti Spaces In Topological Spaces
S. Maragathavalli
Department of Mathematics, Sree Saraswathi Thyagaraja College, Pollachi,
Coimbatore District, Tamil Nadu, India
*smvalli@rediffmail.com
Abstract
In this paper we introduce the concept of γ-sαg*-open sets and discuss some of their basic properties.
Key words: γ-sαg*-semi Ti spaces (γ, β)-sαg*-semi continuous maps.
1. Introduction
The study of semi open set and semi continuity in topological space was initiated by Levine[14].
Bhattacharya and Lahiri[3] introduced the concept of semi generalized closed sets in the topological
spaces analogous to generalized closed gets introduced by Levine[15]. Further they introduced the
semi generalized continuous functions and investigated their properties. Kasahara[11] defined the
concept of an operation on topological spaces and introduced the concept of α-closed graphs of a
function. Jankovic[10] defined the concept of α-closed sets. Ogata [21] introduced the notion of τγ
which is the collection of all γ-open sets in topological space (X, τ) and investigated the relation
between γ-closure and τγ-closure.
We introduce the notion γ-sαg*-semi Ti (I = 0, ½, 1, 2) spaces. In section 4, we introduce (γ,
β)-sαg*-semi continuous map which analogous to (γ, β)-continuous maps and investigate some
important properties. Finally we introduce (γ, β)-sαg*-semi homeomorphism in (X, τ) and study
some of their properties.
2. Premilinaries
Throughout this paper (X, ) represent non-empty topological space on which no separation axioms
are assumed unless otherwise mentioned. For a subset A of a space (X, ), cl(A), int(A) denote the
closure and interior of A respectively. The intersection of all -closed sets containing a subset A of
(X, ) is called the -closure of A and is denoted by cl(A).
2.1 Definition [11]
Let (X, τ) be a topological space. An operation γ on the topology τ is a mapping from τ on to power
set P(X) of X such that V ⊆ Vγ for each V ∈ τ, where Vγ denote the value of γ at V. It is denoted by γ:
τ → P(X).
2.2 Definition [21]
A subset A of a topological space (X, τ) is called γ-open set if for each x ∈ A there exists a open set U
such that x ∈U and Uγ ⊆ A. τγ denotes set of all γ-open sets in (X, τ).
2.3 Definition [21]
The point x ∈ X is in the γ-closure of a set A ⊆ X if Uγ ∩ A ≠ φ for each open set U of x. The
γ-closure of set A is denoted by clγ(A).
2.4 Definition [21]
Let (X, τ) be a topological space and A be subset of X then τγ -l(A) = ∩ {F : A ⊆ F, X – F ∈ τγ }
2.5 Definition [21]
Let (X, τ) be topological space. An operation γ is said to be regular if, for every open neighborhood U
1
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.3, 2012
and V of each x∈X, there exists an open neighborhood W of x such that Wγ ⊆ Uγ ∩ Vγ.
2.6 Definition [21]
A topological space (X, τ) is said to be γ-regular, where γ is an operation of τ, if for each x ∈X and for
each open neighborhood V of x, there exists an open neighborhood U of x such that Uγ contained in V.
2.7 Remark [21]
Let (X, τ) be a topological space, then for any subset A of X, A ⊆ cl(A) ⊆ clγ (A) ⊂ τγ-cl(A).
2.8 Definition [24]
A subset A of (X, τ) is said to be a γ-semi open set if and only if there exists a γ-open set U such that U
⊆ A ⊆ clγ(U).
2.9 Definition [24]
Let A be any subset of X. Then τγ-int (A) is defined as τγ-int (A) = ∪{U:U is a γ-open set and U ⊆ A}
2.10 Definition[24]
A subset A of X is said to be γ-semi closed if and only if X – A is γ-semi open.
2.11 Definition[24]
Let A be a subset of X. There τγ-scl (A) = ∩ {F: F is γ-semi closed and A ⊆ F}.
2.12 Definition[20]
A subset A of (X, τ) is said to be a strongly αg*-closed set if αcl(A) ⊆ U whenever A ⊆ U and U is
g*-open in (X,τ).
2.13 Definition[20]
If a subset A of (X, τ) is a strongly αg*-closed set then X – A is a strongly αg*-open set.
2.14 Definition[20]
A space (X, τ) is said to be a s*Tc-space if every strongly αg*-closed set of (X, τ ) is closed in it.
2.15 Definition [20]
A space (X, τ) is called
(i) a γ-semi To space if for each distinct points x, y ∈ X, there exists a γ-semi open set U such that
x ∈ U and y ∉ U or y ∈ U and x ∉ U.
(ii) a γ-semi T1 space if for each distinct points x, y ∈ X, these exist γ-semi open sets U, V
containing x and y respectively such that y ∉ U and x ∉ V.
(iii) a γ - semi T2 space if for each x, y ∈ X there exists a γ-semi open sets U, V such that x ∈ U and y
∈ V and U ∩ V = φ.
2.16 Definition [24]
A subset A of (X, τ) is said is be γ-semi g-closed if τγ-scl(A) ⊆ U whenever A ⊆ U and U is a
γ-semi open set in (X, τ).
2.17 Definition [24]
A space (X, τ) is said to be γ-semi T1/2-space if every semi g-closed set in (X, τ) is γ-semi closed.
2.18 Definition[24]
A mapping f: (X, τ) → (y, σ) is said to be (γ, β) -semi continuous if for each x of X and each β-semi
open set V containing f(x) there exists a γ-semi open set U such that x ∈ U and f(U) ⊆ V.
2.19 Definition [24]
2
3. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.3, 2012
A mapping f : (X, τ) → (Y, σ) is said to be (γ, β)-semi closed if for any γ-semi closed
set A of (X, τ), f(A) is a β-semi closed.
2.20 Definition [24]
A mapping f : (X, τ) → (Y, σ) is said to be (γ, β)-semi homeomorphism, if f is bijective, (γ,
β)-semi-continuous and f -1 is (β, γ )-semi continuous.
2.21 Definition
A subset A of (X, τ) is said to be a γ-sαg*-semi open set if and only if there exists a γ-sαg*-open
set U such that U⊆ A⊆ cl γ(U).
2.22 Theorem
If A is a γ-semi open set in (X, τ), then A is a γ-sαg*-semi open set.
2.23 Definition
A subset A of X is said to be γ-sαg*-semi closed if and only if X − A is γ-sαg*-semi open.
2.24 Definition
Let A be a subset of X. Then τγs*-scl(A) = ∩ {F : F is γ-sαg* semi closed and A ⊆ F} .
2.25 Theorem
For a point x ∈ X, x ∈ τγs*-scl(A) if and only if V ∩ A ≠ φ for any V ∈ τγs*-SO(X ) such that x ∈ V.
2.26 Remark
From the Theorem 3.12 and the Definition 3.25 we have A ⊆ τγs*-scl(A) ⊆ τγs*-cl(A) for any subset
A of (X, τ).
2.27 Remark
Let γ: τ → P(X ) be a operation. Then a subset A of (X, τ) is γ-sαg*-semi closed if and only if
τγs*-scl(A)=A
3. γ-sαg*-Semi Ti Spaces
α
In this section, we investigate a general operation approaches on Ti spaces where
i = 0, ½, 1,2. Let γ : τ → P(X ) be a operation on a topology τ.
3.1 Definition
A space (X, τ) is called γ-sαg*-semi T0 space if for each distinct points
x, y ∈ X there exists a γ-sαg*-semi open set U such that x ∈ U and y ∉ U or y ∈U and x ∉ U.
3.2 Definition
A space (X, τ) is called γ-sαg* semi T1 space if for each distinct points x, y∈ X there exists γ-sαg*
semi open sets U, V containing x and y respectively such that y ∉ U and x ∉ V.
3.3 Definition
A space (X, τ) is called a γ-sαg*-semi T2 space if for each x, y∈ X there exist γ-sαg*-semi open sets U,
V such that x ∈ U and y ∈ V and U ∩ V = φ.
3.4 Definition
A subset A of (X, τ) is said to be γ-sαg*-semi g-closed if τγ-scl(A) ⊆ U whenever A ⊆ U and U is a
γ-sαg*-semi open set in (X, τ).
3.5 Remark
From Theorem 3.16 and Remark 3.28 we have every γ-sαg*-semi g-closed set is γ-semi g-closed.
3
4. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.3, 2012
3.6 Definition
A space (X, τ) is γ-sαg*-semi T1/2 space if every γ-sαg*-semi g-closed set in (X, τ) is γ-semi closed.
3.7 Remark
Let A be a subset of X. Then τγs*-scl(A) ⊆ τγ-scl (A).
Proof
Let x ∉ τγ-scl(A)
⇒ x ∉ ∩ {F:F is γ - semi closed and A ⊆ F}
⇒ x ∉ F where F is γ - semi closed and A⊆ F
⇒ x ∉ F where F is γ - sαg* -semi closed and A⊆ F
⇒ x ∉ ∩ {F : F is γ - sαg*-semi closed and A⊆ F}
⇒ x ∉ τγs*-scl(A)
Therefore, τγ-scl(A) ⊆ τγs*-scl(A).
3.8 Theorem
A subset A of (X, τ) is γ-sαg*-semi g-closed if and only if τγs*-scl({x}) ∩ A ≠ φ holds for every x ∈ τγ
-scl(A).
Proof
Let U be γ-sαg*-semi open set such that A ⊆ U. Let x ∈ τγ-scl(A). By assumption there exists a z
∈ τγs*-scl({x}) and z ∈ A ⊆ U. It follows from Theorem 3.27 that U ∩ {x} ≠ φ. Hence x ∈ U.
This implies τγ-scl (A) ⊆ U. Therefore, A is γ-sαg*-semi g-closed set in (X, τ).
Conversely, suppose x ∈ τγ-scl(A) such that τγs*-scl({x}) ∩ A = φ. Since
τγs*-scl ({x}) is γ-sαg*-semi closed set in (X, τ), from the Definition 3.24, (τγs*-scl({x})c is a
γ-sαg*-semi open set. Since A ⊆ τγs*-scl({x})c and A is γ-sαg*-semi-g-closed set, we have τγ-scl(A)
⊆ τγs*-scl ({x})c. Hence x ∉ τγ-scl(A) . This is a contradiction. Hence τγs*-scl({x}) ∩ A ≠ φ.
3.9 Theorem
If τγs*-scl({x}) ∩ A ≠ φ holds for every x ∈ τγs*-scl(A), then τγs*-scl(A) − A does not contain a non
empty γ-sαg*-semi closed set.
Proof
Suppose there exists a non empty γ-sαg*-semi closed set F such that F ⊆ τγs*-scl(A) − A. Let x ∈ F, x
∈ τγs*-scl(A) holds. It follows from Remark 3.28 and 3.29, φ ≠ F ∩ A = τγs*-scl(F) ∩ A ⊇ τγs*-scl
({x}) ∩ A which is a contradiction. Thus, τγs*-scl(A) – A does not contains a non empty γ-sαg*-semi
closed set.
3.10 Theorem
Let γ : τ → P(X ) be an operation. Then for each x ∈ X, {x} is γ-sαg*-semi closed or {x} c is
γ-sαg*-semi g-closed set in (X, τ ).
Proof
Suppose that {x} is not γ- sαg*-semi closed then X–{x} is not γ-sαg*-semi open. Let U be any
γ-sαg*-semi open set such that {x}c ⊆ U. Since U = X, we have τγ -scl ({x}) c ⊆ U. Therefore, {x}
c
is a γ-sαg*-semi g-closed set.
3.11Theorem
A space (X, τ) is γ-sαg*-semi-T½ space if and only if {x} is γ-sαg*-semi closed or γ-sαg*- semi
open in (X, τ).
Proof
Suppose {x} is not γ-sαg*-semi closed Then, it follows from assumption and Theorem 3.10, {x} is
γ-sαg*-semi open.
Conversely, Let F be γ-sαg*-semi g-closed set in (X, τ). Let x be any point in
τγs*-scl(F), then {x} is γ-sαg*-semi open or γ-sαg*-semi closed.
Case (i) : Suppose {x} is γ-sαg*-semi open. Then by Theorem 3.27, we have
4
5. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.3, 2012
{x} ∩ F ≠ φ. Hence x ∈ F.
Case (ii): suppose {x} is γ-sαg*-semi closed. Assume x ∉ F, Then x ∈ τγs*-scl(F) – F. This is not
possible by Theorem 3.9. Thus we have x ∈ F. Therefore, τγs*-scl(F) = F and hence F is
γ-sαg*-semi closed.
3.13 Remark
Let X = {a, b, c}, τ = {φ, X, {a}, {b}, {a, b}, {a, c}}, define γ : τ → P(X) be an operation such that for
every A ∈ τ, Aγ = A if b ∈ A, Aγ = cl(A) if b ∉ A. Then (X, τ) is γ- sαg*- semi T0 but it is neither
γ-sαg*-semi T2 nor γ-sαg*-semi T½ nor γ-sαg*-semi T1.
4. (γ, β)-sαg*-SEMI CONTINUOUS MAPS
γ α
Through out this chapter let (X, τ) and (Y, σ) the two topological spaces and let γ : τ → P(X) and β: σ →
P(Y) be operations on τ and σ respectively.
4.1 Definition
A mapping f : (X, τ) → (Y, σ) is said to be (γ, β)-sαg*-semi continuous if for each x of X and each
β-sαg*-semi open set V containing f(x) there exists a γ-sαg*-semi open set U such that x∈U and f (U) ⊆ V.
4.2 Remark
If (X, τ) and (Y, σ) are both γ-sαg*-regular spaces then the concept of (γ, β)-sαg*-semi continuity and semi
continuity are coincide.
4.3 Theorem
Let f: (X, τ) → (Y, σ) be (γ, β)- sαg*-semi continuous mapping. Then,
(i) f (τγs*-scl(A)) ⊆ τβs*-scl (f(A)) holds for every subset A of (X, τ).
(ii) Let γ be an operation, then for every β-sαg*-semi closed set B of (Y, σ), f -1(B) is γ-sαg*-semi closed in
(X, τ)
Proof
(i) Let y ∈ f (τγs*-scl(A)) and V be any β-sαg*-semi open set containing y. Then there exists a point x ∈
X and γ-sαg*-semi open set U such that f(x) = y and x ∈ U and f(U) ⊆ V. Since x ∈ τγs*-scl(A), We have
U ∩ A ≠ φ and hence φ ≠ f (U ∩ A) ⊆ f(U) ∩ f(A) ⊆ V ∩ f(A). This implies f(x) ∈ τβs*-scl(f(A)).
Therefore, we have f (τγs*-scl(A)) ⊆ τβs*-scl(f(A)).
(ii) Let B be a β-sαg*-semi closed set in (Y, σ). Therefore, τβs*-scl(B) = B. By using (i) we have f(τγs*-scl
(f -1(B))) ⊆ τβs*-scl (B) = B. Therefore we have τγs*-scl(f -1(B)) ⊆ (f -1(B)). Hence f -1(B) is γ-sαg*-semi
closed.
4.4 Definition
A mapping f : (X, τ)→(Y, σ) is said to be (γ, β)-sαg*-semi closed if for any γ-sαg*-semi closed set A of
(X, τ), f(A) is a β-sαg*-semi closed .
4.5 Theorem
Suppose that f is (γ, β)-sαg*-semi continuous mapping and f is (γ, β)- sαg*-semi closed. Then for every
γ-sαg*-semi g-closed set A of (X, τ) the image f(A) is β-sαg*-semi-g-closed.
Proof
Let V be any β-sαg*-semi open set in (Y, σ) such that f(A) ⊆ V. By using Theorem 4.3 (ii), f -1(V) is
γ-sαg*-semi open. Since, A is γ-sαg*-semi g-closed and A ⊆ f -1(V), we have τγs*-scl(A) ⊆ f -1(V), and
hence f(τγs*-scl(A)) ⊆ V. It follows from the assumption that f(τγs*-scl(A)) is a β-sαg*-semi closed set.
Therefore, τβs*-scl(f(A))) ⊆ τβs*-scl(f(τγs*-scl(A)) = f(τγs*-scl(A)) ⊆ V. This implies f(A) is
β-sαg*-semi-g-closed.
4.6 Theorem
Let f: (X, τ) → (Y, σ) be (γ, β)-sαg*-semi continuous and (γ, β)-sαg*- semi closed. If f is injective and
5
6. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.3, 2012
(Y, σ) is β-sαg*-semi T½, then (X, τ) is γ-sαg*-semi T½ space.
Proof
Let A be γ-sαg*-semi-g-closed set in (X, τ). Now, to show that A is γ-sαg*-semi closed. By Theorem
4.5, (i) and assumption it is obtained that f(A) is β-sαg*-semi-g-closed and hence f(A) is
β-sαg*-semi-g-closed. By Theorem 5.4(ii), f –1(f(A)) is γ-sαg*-semi closed in (X, τ). Therefore, A is
γ-sαg*-semi closed in (X, τ). Hence (X, τ) is γ-sαg*-semi T½ space.
4.7 Definition
A mapping f : (X, τ) → (Y, σ) said to be (γ, β)-sαg*-semi homeomorphism, if f is bijective,
(γ, β)-sαg*-semi continuous and f -1 is (β, γ)-sαg*-semi continuous.
4.8 Theorem
Let f: (X, τ) → (Y, σ) be (γ, β)-sαg*-semi homeomorphism and (γ, β)-sαg*-semi closed. If (Y, σ) is
β-sαg*-semi T½ then (X, τ) is γ-sαg*-semi T½ space.
Proof
Follows from Theorem 4.5.
4.9 Theorem
Let f : (X, τ) → (Y, σ) be (γ, β)-sαg*-semi continuous injection. If (Y, σ) is β-sαg*-semi T1 (resp. β-
sαg*- semi T2) then (X, τ) is γ-sαg*-semi T1 (resp. γ-sαg*-semi T2).
Proof
Suppose (Y, σ) is β-sαg*-semi T2. Let x and y be distinct points in X. Then, there exists two
γ-sαg*-semi open sets V and W of Y such that f(x ) ∈ V, f(y) ∈ W and V ∩ W = φ. Since f is (γ,
β)-sαg*-semi continuous for V and W there exists two γ-sαg*-semi open set U and S such that x ∈ U, y
∈ S, and f(U) ⊆ V and f(S) ⊆ W. Therefore, U ∩ S = φ. Hence (X, τ) is γ-semi-sαg*-T2 space.
Similarly, we can prove the case β-sαg*-semi T1.
5. Conclusion
The γ-sαg*-open sets, γ-sαg*-semi Ti spaces, (γ, β)-sαg*-semi continuous maps may be used to find
decomposition of γ-sαg*-semi Ti spaces. We can also define separation axioms for the γ-sαg*-semi
Ti spaces.
References
Balachandran, K., Sundaram, P., & Maki, K., (1991), “On generalized continuous maps in topological
spaces”, Mem. Fac. Sci. Kochi Univ. Ser. A. Math, 12, 3.13.
Balasubramanian, G., (1982), “On some generalizations of compact spaces”, Glasnik, Math Ser. III, 17, 367
– 380.
Bhattacharyya, P., & Lahiri,B. K., (1987), “Semi-generalized closed sets in topology”, Indian J. Math., 29,
376 – 382.
Biswas, N., (1970), “On characterizations of semi-continuous functions”, Atti. Accad. Nax Lincei
Rend. cl. Sci. Fis. Math. Atur. (8), 48, 399 – 402.
Crossely, S.G., & Hildebrand, S.K., (1971), “Semi closure”, Texas. J. Sci., 22, 99 – 122.
Crossely, S.G., & Hildebrand, S.K., (1972), “Semi-topological properties”, Fund. Math. 74, 41 – 53.
Devi, R., Maki, H.,& Balachandran, K., (1993), “Semi – generalized closed maps and generalized semi –
closed maps”, Mem. Fac. Sci Kochi Univ Ser. A. Math., 14 41 – 53.
6
7. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.3, 2012
Dunham, W., “T spaces”, (1977), Kyungpook Math J., 17, 161 – 169.
Jankovic, D. S., (1983), “On functions with α-closed graphs”, Glasnik Math., 18, 141 – 148.
Kasahara,S., (1979), “Operation – compact spaces”, Math. Japonica 24, 97 – 103.
Kasahara, s., (1975) , “On weakly compact regular spaces”, II Proc. Japan Acad., 33, 255 – 259.
Kasahara,S., (1973), “Characterization of compactness and countable compactness”, Proc. Japan Acad., 49,
523 – 523.
Levine, N., (1963), “Semi open sets and semi – continuity in topological spaces”, Amer. Math. Monthly,
70, 36 – 41.
Levine, N., (1970), “Generalized closed sets in topology”, Rend. Circ. Math. Palerno, (2) 19 (1970), 89 –
96.
Maki, H., Ogata, H., Balachandran, K., Sundram, P., & Devi, R., (2000), “The digital line operation
approaches of T1/2 space”, Scientiae Mathematicae, 3, 345 – 352.
Maki, H. & Nori,T., “Bioperations and some separation axioms”, Scientiae Mathematicae Japonicae
Online, 4, 165 – 180.
Maki, H. Balachandran K. & Devi,R., (1996), “Remarks on semi-generalized closed sets and generalized
semi–closed sets”, Kungpook Math. J., 36(1), 155 – 163.
Maki, H., Sundram, P. & Balachandran,K., (1991), “semi-generalized continuous maps and
semi-T1/2-spaces”, Bull . Fukuoka Univ. Ed., Part III, 40, 33-40.
Maragatharalli, S. & Shick John, M., (2005), “On strongly αg* - closed sets in topological spaces”, ACTA
CIENCIA INDICA, Vol XXXI 2005 No.3, , 805 - 814.
Ogata, H., (1991), Operation on topological spaces and associated topology, Math Japonica. 36(1), 175 –
183.
Ogata, T., (1991), “Remarks on some operation-separation axioms”, Bull Fukuoka Univ. Ed. Part III, 40,
41– 43.
Noiri,T., (1971), “On semi-continuous mappings”, Atti Accad. Naz. Lincei Rend. cl. Sci. Fis. Math. Natur.
(8) 54, 41 – 43.
Sai Sundara Krishnan, G., “A new class of semi open sets in Topological spaces”, International Journal of
Mathematics and Mathematical Sciences.
L.A. Steen. L. A. & Seebach, J. A. Jr. (1978), “Counter Examples in Topology”, Springer-Verlag. New
York.
Umehara, J. & Maki. H. (1990), “Operator approaches of weakly Hausdroff spaces”, Mem. Fac. Sci. Kochi
Unvi. Ser. A, Math., 11, 65 – 73.
Umehara, J., (1994) “A certain bioperation on topological spaces”, Mem. Fac. Sci. Kochi. Univ. Ser. A,
Math., 15, 41 – 49.
7
8. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.3, 2012
Note 1: From the Definitions, Theorem 3.11 and 3.12 and Remarks 3.13, 4.12 [24] we get
γ-sαg* γ-sαg* γ-sαg* γ-sαg*
semi T2 semi T1 semi T ½ semi T0
γ-semi T0 γ-semi T2 γ-semi T1 γ-semi T ½
γT2 γT1 γT ½ γT0
T2 T1 T½ T0
Where A → B represent A implies B but not conversely.
8