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.
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.
This document introduces and studies properties of strongly wgrα-continuous and perfectly wgrα-continuous functions between topological spaces. It shows that if a function is perfectly wgrα-continuous, then it is also perfectly continuous and strongly wgrα-continuous. If a function is strongly wgrα-continuous and the codomain space is T_wgrα, then the function is also continuous. The composition of two perfectly wgrα-continuous functions is also perfectly wgrα-continuous. The document also introduces wgrα-compact and wgrα-connected spaces and studies some of their properties.
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
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.
Stability criterion of periodic oscillations in a (2)Alexander Decker
This document introduces and investigates the properties of contra ω-quotient functions, contra ω-closed functions, and contra ω-open functions using ω-closed sets. It defines these types of functions and explores their basic properties and relationships. Some examples are provided to illustrate that the composition of contra ω-closed mappings is not always contra ω-closed. Several theorems are also presented regarding the compositions of these types of mappings.
In this paper, the concepts of wgr?-I-closed maps, wgr?-I-homeomorphism, wgr?-I-connectedness and wgr?-I-compactness are introduced and some their properties in ideal topological spaces are investigated.
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.
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.
This document introduces and studies properties of strongly wgrα-continuous and perfectly wgrα-continuous functions between topological spaces. It shows that if a function is perfectly wgrα-continuous, then it is also perfectly continuous and strongly wgrα-continuous. If a function is strongly wgrα-continuous and the codomain space is T_wgrα, then the function is also continuous. The composition of two perfectly wgrα-continuous functions is also perfectly wgrα-continuous. The document also introduces wgrα-compact and wgrα-connected spaces and studies some of their properties.
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
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.
Stability criterion of periodic oscillations in a (2)Alexander Decker
This document introduces and investigates the properties of contra ω-quotient functions, contra ω-closed functions, and contra ω-open functions using ω-closed sets. It defines these types of functions and explores their basic properties and relationships. Some examples are provided to illustrate that the composition of contra ω-closed mappings is not always contra ω-closed. Several theorems are also presented regarding the compositions of these types of mappings.
In this paper, the concepts of wgr?-I-closed maps, wgr?-I-homeomorphism, wgr?-I-connectedness and wgr?-I-compactness are introduced and some their properties in ideal topological spaces are investigated.
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.
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 Some Continuous and Irresolute Maps In Ideal Topological Spacesiosrjce
In this paper we introduce some continuous and irresolute maps called
δ
ˆ
-continuity,
δ
ˆ
-irresolute,
δ
ˆ
s-continuity and
δ
ˆ
s-irresolute maps in ideal topological spaces and study some of their properties.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
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.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
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.
Research Inventy : International Journal of Engineering and Scienceinventy
esearch 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.
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.
Some forms of N-closed Maps in supra Topological spacesIOSR Journals
In this paper, we introduce the concept of N-closed maps and we obtain the basic properties and
their relationships with other forms of N-closed maps in supra topological spaces.
(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.
This document discusses topological gα-WG quotient mappings. It begins by introducing gα-WG closed sets and defines a gα-WG quotient map using these sets. It studies the basic properties of gα-WG quotient maps and their relationships to other topological mappings such as gα-quotient maps. Examples are provided to illustrate the concepts. The document provides relevant definitions and preliminaries on topological concepts such as α-open sets, w-closed sets, and different types of continuous mappings. It then defines gα-WG quotient maps and strongly gα-WG quotient maps and establishes properties and relationships between these mappings.
On the-approximate-solution-of-a-nonlinear-singular-integral-equationCemal Ardil
This document summarizes a study on finding approximate solutions to nonlinear singular integral equations. The study proves the existence and uniqueness of solutions to such equations defined on bounded regions of the complex plane. It then presents a method for finding approximate solutions using an iterative fixed-point principle approach. Nonlinear singular integral equations have many applications in fields like elasticity, fluid mechanics, and mathematical physics. The study contributes to improving methods for solving these important types of equations.
Contra * Continuous Functions in Topological SpacesIJMER
This document discusses contra α* continuous functions between topological spaces. It begins by introducing α*-open sets and various related concepts like α*-continuity. It then defines a function from one topological space to another to be contra α*-continuous if the preimage of every open set is α*-closed in the domain space. Some properties of contra α*-continuous functions are established, including that every contra-continuous function is contra α*-continuous. Examples are given to show the concepts are independent. The discussion considers the relationships between contra α*-continuity and other variations of contra-continuity.
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 (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.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
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.
δ ˆ – Closed Sets in Ideal Topological SpacesIOSR Journals
The document introduces the concept of δˆ-closed sets in ideal topological spaces. It defines a subset A to be δˆ-closed if the σ-closure of A is contained in every open set U containing A. Some basic properties of δˆ-closed sets are established, including that δ-closed, δ-I-closed, δg-closed, and δgˆ-closed sets are all δˆ-closed. However, the converse relationships are not always true. Examples are provided to illustrate the independence of these classes of closed sets.
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
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.
This document introduces and studies the concepts of πgr-homeomorphisms and πgrc-homeomorphisms between topological spaces. It begins by providing definitions of related concepts such as πgr-closed maps, πgr-continuous maps, and πgr-irresolute maps. It then defines πgr-homeomorphisms as bijections that are both πgr-continuous and πgr-open, and πgrc-homeomorphisms as bijections whose inverse images are πgr-closed sets. Several properties and characterizations of these maps are established. It is shown that πgr-homeomorphisms and πgrc-homeomorphisms
The document defines and studies the properties of g#p-continuous maps between topological spaces. It is shown that:
1. Every pre-continuous, α-continuous, gα-continuous and continuous map is g#p-continuous.
2. The class of g#p-continuous maps properly contains and is properly contained in other classes of generalized continuous maps.
3. g#p-continuity is independent of other properties like semi-continuity and β-continuity.
4. The composition of two g#p-continuous maps need not be g#p-continuous.
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.
This document introduces and investigates the concept of contra-#rg-continuous functions between topological spaces. It defines contra-#rg-continuity and related concepts like contra-#rg-irresolute functions. Several properties of contra-#rg-continuous functions are proven, including that every contra-continuous function is contra-#rg-continuous, and the composition of a contra-#rg-continuous function with a continuous function is contra-#rg-continuous. Examples are provided to show certain concepts like contra-#rg-continuity and #rg-continuity are independent. The relationship between contra-#rg-continuity and other types of generalized continuous functions is also examined.
This document summarizes a research paper that studied the class of β-normal spaces. β-normal spaces generalize p-normal and s-normal spaces. The paper investigates the relationships between these classes of spaces and properties of β-normal spaces. It also studies various forms of generalized β-closed functions and their properties. Key results shown include that a space is β-normal if and only if it satisfies two equivalent properties and that if a function is β-closed and continuous and the domain space is normal, then the range space is β-normal. Diagrams of implications between the different classes of spaces and types of functions are also presented.
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 Some Continuous and Irresolute Maps In Ideal Topological Spacesiosrjce
In this paper we introduce some continuous and irresolute maps called
δ
ˆ
-continuity,
δ
ˆ
-irresolute,
δ
ˆ
s-continuity and
δ
ˆ
s-irresolute maps in ideal topological spaces and study some of their properties.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
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.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
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.
Research Inventy : International Journal of Engineering and Scienceinventy
esearch 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.
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.
Some forms of N-closed Maps in supra Topological spacesIOSR Journals
In this paper, we introduce the concept of N-closed maps and we obtain the basic properties and
their relationships with other forms of N-closed maps in supra topological spaces.
(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.
This document discusses topological gα-WG quotient mappings. It begins by introducing gα-WG closed sets and defines a gα-WG quotient map using these sets. It studies the basic properties of gα-WG quotient maps and their relationships to other topological mappings such as gα-quotient maps. Examples are provided to illustrate the concepts. The document provides relevant definitions and preliminaries on topological concepts such as α-open sets, w-closed sets, and different types of continuous mappings. It then defines gα-WG quotient maps and strongly gα-WG quotient maps and establishes properties and relationships between these mappings.
On the-approximate-solution-of-a-nonlinear-singular-integral-equationCemal Ardil
This document summarizes a study on finding approximate solutions to nonlinear singular integral equations. The study proves the existence and uniqueness of solutions to such equations defined on bounded regions of the complex plane. It then presents a method for finding approximate solutions using an iterative fixed-point principle approach. Nonlinear singular integral equations have many applications in fields like elasticity, fluid mechanics, and mathematical physics. The study contributes to improving methods for solving these important types of equations.
Contra * Continuous Functions in Topological SpacesIJMER
This document discusses contra α* continuous functions between topological spaces. It begins by introducing α*-open sets and various related concepts like α*-continuity. It then defines a function from one topological space to another to be contra α*-continuous if the preimage of every open set is α*-closed in the domain space. Some properties of contra α*-continuous functions are established, including that every contra-continuous function is contra α*-continuous. Examples are given to show the concepts are independent. The discussion considers the relationships between contra α*-continuity and other variations of contra-continuity.
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 (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.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
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.
δ ˆ – Closed Sets in Ideal Topological SpacesIOSR Journals
The document introduces the concept of δˆ-closed sets in ideal topological spaces. It defines a subset A to be δˆ-closed if the σ-closure of A is contained in every open set U containing A. Some basic properties of δˆ-closed sets are established, including that δ-closed, δ-I-closed, δg-closed, and δgˆ-closed sets are all δˆ-closed. However, the converse relationships are not always true. Examples are provided to illustrate the independence of these classes of closed sets.
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
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.
This document introduces and studies the concepts of πgr-homeomorphisms and πgrc-homeomorphisms between topological spaces. It begins by providing definitions of related concepts such as πgr-closed maps, πgr-continuous maps, and πgr-irresolute maps. It then defines πgr-homeomorphisms as bijections that are both πgr-continuous and πgr-open, and πgrc-homeomorphisms as bijections whose inverse images are πgr-closed sets. Several properties and characterizations of these maps are established. It is shown that πgr-homeomorphisms and πgrc-homeomorphisms
The document defines and studies the properties of g#p-continuous maps between topological spaces. It is shown that:
1. Every pre-continuous, α-continuous, gα-continuous and continuous map is g#p-continuous.
2. The class of g#p-continuous maps properly contains and is properly contained in other classes of generalized continuous maps.
3. g#p-continuity is independent of other properties like semi-continuity and β-continuity.
4. The composition of two g#p-continuous maps need not be g#p-continuous.
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.
This document introduces and investigates the concept of contra-#rg-continuous functions between topological spaces. It defines contra-#rg-continuity and related concepts like contra-#rg-irresolute functions. Several properties of contra-#rg-continuous functions are proven, including that every contra-continuous function is contra-#rg-continuous, and the composition of a contra-#rg-continuous function with a continuous function is contra-#rg-continuous. Examples are provided to show certain concepts like contra-#rg-continuity and #rg-continuity are independent. The relationship between contra-#rg-continuity and other types of generalized continuous functions is also examined.
This document summarizes a research paper that studied the class of β-normal spaces. β-normal spaces generalize p-normal and s-normal spaces. The paper investigates the relationships between these classes of spaces and properties of β-normal spaces. It also studies various forms of generalized β-closed functions and their properties. Key results shown include that a space is β-normal if and only if it satisfies two equivalent properties and that if a function is β-closed and continuous and the domain space is normal, then the range space is β-normal. Diagrams of implications between the different classes of spaces and types of functions are also presented.
This document summarizes a research paper that studied the class of β-normal spaces. β-normal spaces generalize p-normal and s-normal spaces. The paper investigates the relationships between these classes of spaces and properties of β-normal spaces. It also studies various forms of generalized β-closed functions and their properties. Key results shown are that the implications in normality hold for β-normal, p-normal, and s-normal spaces, and properties characterizing β-normal spaces. The paper defines concepts like β-closed sets and β-neighborhoods that are used to study β-normality and generalized β-closed functions.
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.
11. gamma sag semi ti spaces in topological spacesAlexander Decker
This document introduces the concept of γ-sαg*-semi Ti spaces where i = 0, 1/2, 1, 2. It defines γ-sαg*-semi open and closed sets. Properties of γ-sαg*-semi closure and γ-sαg*-semi generalized closed sets are discussed. It is shown that every γ-sαg*-semi generalized closed set is γ-semi generalized closed. A subset A is γ-sαg*-semi generalized closed if and only if the intersection of A with the γ-sαg*-semi closure of each point in the γ-closure of A is non-empty. The γ-sαg*-semi closure of a set
This document introduces the concept of γ-sαg*-semi Ti spaces where i = 0, 1/2, 1, 2. It defines γ-sαg*-semi open and closed sets. Properties of γ-sαg*-semi closure and γ-sαg*-semi generalized closed sets are discussed. It is shown that every γ-sαg*-semi generalized closed set is γ-semi generalized closed. The paper investigates when a space is a γ-sαg*-semi Ti space by looking at when γ-sαg*-semi generalized closed sets are γ-semi closed. It concludes that for each point x in a space, the singleton {x} is either γ-
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
The authors Selvi.R, Thangavelu.P and
Anitha.m introduced the concept of
-continuity between a
topological space and a non empty set where
{L, M, R, S}
[4]. Navpreet singh Noorie and Rajni Bala[3] introduced the
concept of f#
function to characterize the closed, open and
continuous functions. In this paper, the concept of Semi- -
continuity is introduced and its properties are investigated and
Semi- -continuity is further characterized by using f#
functions
This document defines and provides examples of continuous functions between topological spaces. It can be summarized as follows:
1) A function f from a topological space X to a topological space Y is continuous if the preimage of every open set in Y is open in X.
2) Examples of continuous functions include identity functions, constant functions, and compositions of continuous functions.
3) A function from a space X to a product space Y×Z is continuous if and only if its coordinate projections to Y and Z are both continuous.
This document introduces and studies the concept of ˆ-closed sets in topological spaces. Some key points:
1. ˆ-closed sets are defined as sets whose δ-closure is contained in any semi-open set containing the set.
2. It is shown that ˆ-closed sets lie between δ-closed sets and various other classes like δg-closed and ω-closed sets.
3. Several characterizations of ˆ-closed sets are provided in terms of properties of the difference between the δ-closure of the set and the set itself.
4. The concept of the ˆ-kernel of a set is introduced, defined as the intersection of all ˆ-
This research statement summarizes Susovan Pal's postdoctoral research in two areas: 1) Regularity and asymptotic conformality of quasiconformal minimal Lagrangian diffeomorphic extensions of quasisymmetric circle homeomorphisms. This focuses on proving these extensions are asymptotically conformal if the boundary maps are symmetric. 2) Discrete geometry of left conformally natural homeomorphisms of the unit disk from a discrete viewpoint. This constructs homeomorphisms between polygons in the disk that preserve a weighted minimal distance property. The goal is to show these homeomorphisms converge to a continuous one.
This document introduces the concept of fuzzy compact-open topology. Some key points:
- The fuzzy compact-open topology is defined on the class of fuzzy continuous functions between two fuzzy topological spaces.
- An evaluation map from the product of this function class and the domain space into the range space is shown to be fuzzy continuous.
- For fuzzy locally compact Hausdorff domain and range spaces, there is an exponential law isomorphism between the function class with the compact-open topology and the product of this class with the domain space.
This document introduces and investigates some weak separation axioms using the notion of πgb-closed sets. It defines πgb-closed sets, πgb-continuous functions, and various separation axioms including πgb-T0, πgb-T1, and πgb-T2. It introduces the concept of a πgb-D-set and defines associated properties like πgb-D0, πgb-D1, and πgb-D2 spaces. Results are proved relating these new concepts, showing properties like πgb-D1 spaces being πgb-T0 and πgb-D2 spaces being equivalent to
Some properties of gi closed sets in topological space.docxAlexander Decker
This document introduces generalized *i-closed (g*i-closed) sets in topological spaces and studies some of their properties. It defines what a g*i-closed set is and shows that every closed, i-closed, semi-closed, g-closed, gs-closed, and δg-closed set is also a g*i-closed set. However, the converses of these statements are not always true. Examples are provided to illustrate this. The relationships between g*i-closed sets and other generalized closed sets are also examined.
Let's Integrate MuleSoft RPA, COMPOSER, APM with AWS IDP along with Slackshyamraj55
Discover the seamless integration of RPA (Robotic Process Automation), COMPOSER, and APM with AWS IDP enhanced with Slack notifications. Explore how these technologies converge to streamline workflows, optimize performance, and ensure secure access, all while leveraging the power of AWS IDP and real-time communication via Slack notifications.
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
Goodbye Windows 11: Make Way for Nitrux Linux 3.5.0!SOFTTECHHUB
As the digital landscape continually evolves, operating systems play a critical role in shaping user experiences and productivity. The launch of Nitrux Linux 3.5.0 marks a significant milestone, offering a robust alternative to traditional systems such as Windows 11. This article delves into the essence of Nitrux Linux 3.5.0, exploring its unique features, advantages, and how it stands as a compelling choice for both casual users and tech enthusiasts.
Climate Impact of Software Testing at Nordic Testing DaysKari Kakkonen
My slides at Nordic Testing Days 6.6.2024
Climate impact / sustainability of software testing discussed on the talk. ICT and testing must carry their part of global responsibility to help with the climat warming. We can minimize the carbon footprint but we can also have a carbon handprint, a positive impact on the climate. Quality characteristics can be added with sustainability, and then measured continuously. Test environments can be used less, and in smaller scale and on demand. Test techniques can be used in optimizing or minimizing number of tests. Test automation can be used to speed up testing.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
GraphSummit Singapore | The Future of Agility: Supercharging Digital Transfor...Neo4j
Leonard Jayamohan, Partner & Generative AI Lead, Deloitte
This keynote will reveal how Deloitte leverages Neo4j’s graph power for groundbreaking digital twin solutions, achieving a staggering 100x performance boost. Discover the essential role knowledge graphs play in successful generative AI implementations. Plus, get an exclusive look at an innovative Neo4j + Generative AI solution Deloitte is developing in-house.
GraphRAG for Life Science to increase LLM accuracyTomaz Bratanic
GraphRAG for life science domain, where you retriever information from biomedical knowledge graphs using LLMs to increase the accuracy and performance of generated answers
Communications Mining Series - Zero to Hero - Session 1DianaGray10
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
Dr. Sean Tan, Head of Data Science, Changi Airport Group
Discover how Changi Airport Group (CAG) leverages graph technologies and generative AI to revolutionize their search capabilities. This session delves into the unique search needs of CAG’s diverse passengers and customers, showcasing how graph data structures enhance the accuracy and relevance of AI-generated search results, mitigating the risk of “hallucinations” and improving the overall customer journey.
Full-RAG: A modern architecture for hyper-personalizationZilliz
Mike Del Balso, CEO & Co-Founder at Tecton, presents "Full RAG," a novel approach to AI recommendation systems, aiming to push beyond the limitations of traditional models through a deep integration of contextual insights and real-time data, leveraging the Retrieval-Augmented Generation architecture. This talk will outline Full RAG's potential to significantly enhance personalization, address engineering challenges such as data management and model training, and introduce data enrichment with reranking as a key solution. Attendees will gain crucial insights into the importance of hyperpersonalization in AI, the capabilities of Full RAG for advanced personalization, and strategies for managing complex data integrations for deploying cutting-edge AI solutions.
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
available on those devices, but many of the features provide convenience and capability but sacrifice security. This best practices guide outlines steps the users can take to better protect personal devices and information.
AI 101: An Introduction to the Basics and Impact of Artificial IntelligenceIndexBug
Imagine a world where machines not only perform tasks but also learn, adapt, and make decisions. This is the promise of Artificial Intelligence (AI), a technology that's not just enhancing our lives but revolutionizing entire industries.
Programming Foundation Models with DSPy - Meetup SlidesZilliz
Prompting language models is hard, while programming language models is easy. In this talk, I will discuss the state-of-the-art framework DSPy for programming foundation models with its powerful optimizers and runtime constraint system.
Cosa hanno in comune un mattoncino Lego e la backdoor XZ?Speck&Tech
ABSTRACT: A prima vista, un mattoncino Lego e la backdoor XZ potrebbero avere in comune il fatto di essere entrambi blocchi di costruzione, o dipendenze di progetti creativi e software. La realtà è che un mattoncino Lego e il caso della backdoor XZ hanno molto di più di tutto ciò in comune.
Partecipate alla presentazione per immergervi in una storia di interoperabilità, standard e formati aperti, per poi discutere del ruolo importante che i contributori hanno in una comunità open source sostenibile.
BIO: Sostenitrice del software libero e dei formati standard e aperti. È stata un membro attivo dei progetti Fedora e openSUSE e ha co-fondato l'Associazione LibreItalia dove è stata coinvolta in diversi eventi, migrazioni e formazione relativi a LibreOffice. In precedenza ha lavorato a migrazioni e corsi di formazione su LibreOffice per diverse amministrazioni pubbliche e privati. Da gennaio 2020 lavora in SUSE come Software Release Engineer per Uyuni e SUSE Manager e quando non segue la sua passione per i computer e per Geeko coltiva la sua curiosità per l'astronomia (da cui deriva il suo nickname deneb_alpha).
Cosa hanno in comune un mattoncino Lego e la backdoor XZ?
C027011018
1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 2 Issue 7 || July. 2014 || PP-11-18
www.ijmsi.org 11 | P a g e
On πgr - Homeomorphisms in Topological Spaces.
1,
Janaki.C , 2,
Jeyanthi.V
1.
Asst. Professor/Mathematics, L.R.G. Govt.Arts College for Women, Tirupur- 4.
2.
Asst.Professor/Mathematics, Sree Narayana Guru College , Coimbatore – 105.
E-mail: 1.janakicsekar@yahoo.com 2.jeyanthi_sngc@yahoo.com
ABSTRACT:The purpose of this paper is to introduce and study the concept of πgr -closed maps , πgr -
homeomorphism ,πgrc - homeomorphism and obtain some of their characterizations.
KEYWORDS: πgr-closed map, πgr-open map,πgr-homeomorphism and πgrc- homeomorphism.
Mathematics subject classification: 54A05, 54D10.
I. INTRODUCTION
Levine [9]introduced the concept of generalized closed sets in topological spaces and a class of
topological space called T1/2-space. The concept of -closed sets in topological spaces was initiated by
Zaitsav[18] and the concept of g-closed set was introduced by Noiri and Dontchev[4]. N.Palaniappan[16]
studied and introduced regular closed sets in topological spaces. Generalized closed mappings, wg-closed maps
,regular closed maps and rg-closed maps were introduced and studied by Malghan[13],Nagaveni[14],Long[11]
and Arokiarani[1] respectively.Maki et al [12] who introduced generalized homeomorphism and gC-
homeomorphism which are nothing but the generalizations of homeomorphism in topological spaces. Devi et al
[3]defined and studied generalized semi-homeomorphism and gsc homeomorphism in topological spaces. In
2013,Jeyanthi.V and Janaki.C [6] introduced and studied the properties of πgr-closed sets in topological spaces.
Here we introduce and study the concepts of πgr- homeomorphisms ,πgrc -homeomorphism and their relations.
II. PRELIMINARIES
Throughout this paper, X , Y and Z denote the topological spaces (X,τ),(Y,σ) and (Z,η) respectively, on
which no separation axioms are assumed. Let us recall the following definitions.
Definition:2.1
A subset A of a topological space X is said to be
[1] a semi -open [10] if A cl (int(A)) and semi-closed if int (cl(A)) A
[2] a regular open[16] if A = int (cl(A)) and regular closed if A = cl(int(A))
[3] π- open [18] if A is the finite union of regular open sets and the complement of π- open set is π- closed set
in X.
The family of all open sets [ regular open, π-open, semi open] sets of X will be denoted by O(X)(resp. RO(X),
πO(X), SO(X)]
Definition:2.2
A map f: X→Y is said to be
[1] continuous [10]if f-1
(V) is closed in X for every closed set V in Y.
[2] Regular continuous ( r-continuous) [16]if f-1
(V) is regular-closed in X for every closed set
[3] V in Y.
[4] An R-map[2] if f-1
(V) is regular closed in X for every regular closed set V of Y.
[5] πgr-continuous[7,8] if f-1
(V) is πgr-closed in X for every closed set V in Y.
[6] πgr-irresolute[7,8] if f-1
(V) is πgr-closed in X for every πgr -closed set V in Y.
Definition :2.3
A space X is called a πgr-T1/2 space [7,8]if every πgr-closed set is regular closed.
Definition:2.4
A map f: XY is called
1.closed [13 ]if f(U) is closed in Y for every closed set U of X.
2. On Πgr - Homeomorphisms In…
www.ijmsi.org 12 | P a g e
2.almost closed[ 17] if f(U) is closed in Y for every regular closed set U of X.
3.regular closed [11]if f(U) is regular closed in Y for every closed set U of X
4.rc-preserving [15]if f(U) is regular closed in Y for every regular closed set U of X.
Definition:2.5[6]
Let f : (X,τ)→(Y,σ) be a map. A map f is said to be
[1] πgr -open if f(U) in πgr-open in Y for every open set U of X.
[2] strongly gr-open map (M-gr-open)if f(V) is gr-open in Y for every gr-open set V in X.
[3] quasi gr-open if f(V) is open in Y for every gr-open set V in X.
[4] almost gr-open map if f(V) is gr-open in Y for every regular open set V in X.
Definition:2.6
A bijection f:X→ Y is called a homeomorphism [12]if f is both continuous and open.((i.e), f & f-1
are
continuous)
III. πGR - HOMEOMORPHISMS
Definition:3.1
A bijection f:X→ Y is called
[1] πgr - homeomorphism if f is both πgr- continuous and πgr - open.((i.e), f & f-1
are πgr -continuous)
[2] πgrc - homeomorphism if f and f-1
are πgr- irresolute.
Proposition :3.2
If a mapping f : X → Y is πgr -closed, then for every subset A of X, πgr- cl f(A) f(cl(A))
Proof:
Suppose f is πgr -closed and let A X .Then f(cl(A)) is πgr - closed in (Y, ). We have f(A) f(cl(A)). Then
πgr -cl(f(A)) πgr -cl [f(cl(A))] = f(cl(A))
πgr -cl (f(A)) f(cl(A))
Theorem :3.3
Let f : X → Y and g : Y → Z be two mappings such that their composition g f : X → Y be a πgr - closed
map.Then
[1] f is continuous and surjective, then g is πgr- closed.
[2] g is πgr- irresolute and injective, then f is πgr - closed.
[3] f is πgr- continuous, surjective and X is a πgr-T1/2- space, then g is πgr - closed.
Proof :
(i)Let V be a closed set of Y. Since f is Continuous, f-1
(V) is closed in X. Since (g f) is πgr -closed in Z, (g f)
(f-1
(V)) is πgr- closed in Z.
g(f(f-1
(V)) = g(V) is πgr - closed in Z.(Since f is surjective)
ie, for the closed set V of Y, g(V) is πgr- closed in Z.
g is a πgr - closed map.
(ii)Let V be a closed set of X .Since (g f) is πgr - closed ,(g f) (V) is πgr- closed in Z. Since g is πgr -
irresolute,g-1
[(g f)(V)] is πgr - closed in Y.
g-1
[g(f(V))] is πgr closed in Y
f(V) is πgr - closed in Y. Hence f is a πgr - closed map.
(iii)Let V be a closed set of Y
Since f is πgr - continuous, f-1
(V) is πgr - closed in X for every closed set V of Y.Since X is πgr -T1/2- space,
f-1
(V) is regular closed in X and hence closed in X. Now, as in (i), g is a πgr- closed map.
(iv)Let V be a closed set of Y
Since f is πgr - continuous, f-1
(V) is πgr- closed in X.
Since X is πgr -T1/2- space, f-1
(V) is regular closed in X and hence closed in X.
Now, the proof as in (i), g is a πgr- closed map.
3. On Πgr - Homeomorphisms In…
www.ijmsi.org 13 | P a g e
Proposition :3.4
Let f : X → Y and g : Y → Z be πgr - closed maps and Y is a πgr-T1/2- space, then their composition g f :
X→ Z is a πgr - closed map.
Proof :
Let f : X → Y be a closed map. Then for the closed set V of X, f(V) is πgr - closed in Y. Since Y is a πgr -T1/2
space, f(V) is regular closed in Y and hence closed in Y.Again, since g is a πgr - closed map, g(f(V)) is πgr -
closed in Z for the closed set f(V) of Y .
(gf) (V) is πgr - closed in Z for the closed set V of X .
(gf) is a πgr -closed map.
Proposition :3.5
Let f : (X, τ) → (Y, ) be a closed map and g : (Y, ) → (Z,η ) be a πgr - closed map, then their composition
g f : (X, τ) →(Z,η ) is πgr - continuous.
Proof :Let V be a closed set of X. Since f is a closed map, f (V) is closed in Y.
Again, since g is a πgr - closed map, g(f(V)) is a πgr - closed in Z .
(gf) (V) is πgr - closed in Z for the closed set V of X .
(gf) is πgr - closed map.
Proposition:3.6
Let f : X →Y be a πgr- closed map, g : Y → Z be a closed map, Y is πgr -T1/2- space, then their composition
(gf) is a closed map.
Proof :
Let V be a closed set of X. Since f is a πgr - closed map, f(V) is πgr - closed in Y for every closed set V of X.
Since Y is a πgr -T1/2- space, f(V) is regular closed hence closed in Y.Since g is a closed map, then g(f(V)) is
closed in Z.
(gf) (V) is closed in Z for every closed set V of X and hence (g f) is a closed map.
Remark:3.7
a)Homeomorphism and πgr -homeomorphism are independent concepts.
b)Homeomorphism and πgrc -homeomorphism are independent concepts.
Example:3.8
(For both (a) and (b))
(i)Let X= { a,b,c}=Y,τ = { , X, {b} {b,c} {a,b}},σ = { ,Y,{a},{b},{a,b},{a,c}}.Let f : X→ Y be an identity
map. Here the inverse image of open subsets in Y are πgr-open in X and for every open set U of X, f(U) is πgr-
open in Y. Hence f is a πgr - homeomorphism .Also,f and f-1
are πgr-irresolute and hence f is a πgrc-
homeomorphism.
But inverse image of open subsets in Y are not open in X and inverse image of open set U in X is not open in Y.
Hence f is not a homeomorphism . Thus πgr-homeomorphism and πgrc-homeomorphism need not be a
homeomorphism.
(ii)Let X={a,b,c,d}=Y, τ ={ ,X,{c},{d},{c,d},{b,d},{a,c,d},{b,c,d},σ ={ ,Y,{a},{d},{a,d}, {c,d},{
a,c,d},{a,b,d}}.Let f : X → Y be defined by f(a) = b, f(b) = c, f(c) =a, f(d) = d. Here the inverse image of open
sets in (Y, ) are open in (X, ) and the image of open sets in X are open in Y. Hence f is a homeomorphism
.But the inverse image of open sets in (Y, ) are not πgr-open in (X, ) and also the image of open sets in X are
not πgr-open in Y. Hence f is not a πgr - homeomorphism . Also, here f and f-1
are not πgr-irresolute and hence
not a πgrc-homeomorphism.
Remark:3.9
The concepts of πgrc - homeomorphism and πgr- homeomorphism are independent.
Example:3.10
a)Let X ={a,b,c}=Y,τ = { ,X,{b},{a,b}},σ={ ,Y,{b}}.Let f : X Y be an identity map.
4. On Πgr - Homeomorphisms In…
www.ijmsi.org 14 | P a g e
Here the both f and f-1
are πgr- irresolute and not πgr –continuous . Hence πgrc - homeomorphism need not be a
πgr -homeomorphism.
b) Let X ={a,b,c}=Y, τ = { ,X,{a},{b},{a,b}},σ={ ,Y,{b}}. Let f : X Y be an identity map. Here the both
f and f-1
are πgr- continuous and not πgr- irresolute . Hence πgr -homeomorphism need not be a πgrc-
homeomorphism.
The above discussions are summarized in the following diagram:
Homeomorphism
πgr-homeomorphism πgrc-homeomorphism
Remark :3.11
We say the spaces (X,τ) and (Y,) are πgr -homeomorphic (πgrc-homeomorphic)if there exists a πgr-
homeomorphism(πgrc- homeomorphism) from (X, τ) onto (Y, ) respectively . The family of all πgr-
homeomorphism and πgrc-homeomorphisms are denoted by πgrh(X, τ) and πgrch(X, τ).
Proposition :3.12
For any bijection f : (X, τ) →(Y, ), the following statements are equivalent.
[1] f is a πgr - open map
[2] f is a πgr - closed map
[3] f-1
: Y → X is πgr - continuous .
Proof :
(i) (ii) :- Let f be a πgr - open map. Let U be a closed set in X. Then X – U is open in X
By assumption, f(X - U) is πgr - open in Y.
ie, Y – f(X – U) = f(U) is πgr - closed in Y. ie, for a closed set U in X, f(U) is πgr - closed in Y.Hence f is a
πgr - closed map.
(ii) (i) :- let V be a closed set in X .By (ii), f(V) is πgr - closed in Y and f(V) = (f-1
)-1
(V)
f-1
(V) is πgr - closed in Y for the closed set V in Y
f-1
is πgr -continuous.
(iii) (ii) :- let V be open in X .By (iii), (f-1
)-1
(V) = f(V) ie, f(V) in πgr - open in Y
Hence f is a πgr -open map.
Proposition :3.13
Let f : X→Y be a bijective πgr- continuous map. Then the following are equivalent.
[1] f is a πgr -open map.
[2] f is a πgr- homeomorphism.
[3] f is a πgr - closed map.
(i) (iii) also (iii) (i)
f is a πgr- closed map f-1
is πgr - continuous.
Then by part (i) and by the above argument together implies f is a homeomorphism and hence (ii) holds.
Proposition : 3.14
For any bijection f : (X, τ)→(Y, ) the following statements are equivalent.
[1] f-1
: Y →X is πgr - irresolute .
5. On Πgr - Homeomorphisms In…
www.ijmsi.org 15 | P a g e
[2] f is an M- πgr- open map .
[3] f is a M- πgr- closed map.
Proof : (i) (ii) :Let U be a πgr - open set in Y .
By (i), (f-1
)-1
(U) = f(U) is πgr - open in Y.
ie, For the πgr - open set U, f(U) is πgr- open in Y
f is an M - πgr -open map.
(ii) (iii): Let f be an M- πgr- open map
let V be πgr - closed set in X .Then X – V is πgr - open in X. Since f is an - πgr- open map, f(X – V) is πgr-
open in Y..
ie, f(X – V) = Y – f(V) is πgr - open in Y .ie, f(V) is πgr-closed in Y and hence f is an M- πgr- closed map.
(iii) (i): let V be πgr- closed in X .By (iii), f(V) is πgr - closed in Y.Since f-1
is Y X be a mapping and is a
bijection. Again we say that for f(V), πgr - closed in Y, its inverse image (f-1
)-1
(V) is πgr - closed in Y.Hence
f-1
is πgr- irresolute .
Remark : 3.15
Composition of two πgr -homeomorphisms need not be a πgr -homeomorphism.
Example:3.16
Let X = Y = Z={a,b,c},τ={ ,X,{a},{b},{a,b}},σ = { ,Y,{a},{a,b}},η={ ,Z,{c}}. Let us define the mapping
f: X→Y by f(a) = b, f(b) = a, f(c) = c and g:Y→Z by g(a) = b,g(b) = a,
g(c) = c. Here f and g are πgr-homeomorphisms but (g f) is not πgr-continuous and not πgr-open.
ie, (gof)-1
{c} = {c} is not πgr -open in X
Hence composition of two πgr - homeomorphism is not always be a πgr-homeomorphism.
Theorem:3.17
The composition of two πgrc - homeomorphism is a πgrc-homeomorphism .
Proof :
let f : (X, τ) →(Y, ) and g : (Y, ) → (Z,η) be two πgrc - homeomorphic functions.
Let F be a πgr - closed set in Z. Since g is a πgr - irresolute map, g-1
(F) is πgr - closed in (Y, ).Since f is a
πgr - irresolute map, f-1
(g-1
(F)) is πgr - closed in X.
(gof)-1
(F) is πgr - closed in X
(gof) is πgr - irresolute.
Let G be a πgr - closed set in (X,τ).Since f-1
is πgr - irresolute, (f-1
)-1
(G) is πgr - closed in (Y, ).ie, f(G) is πgr
- closed in (Y, )
Since g-1
is πgr - irresolute, (g-1
)-1
(f(G)) = g(f(G)) is πgr- closed in Z
g(f(G)) = (gof) (G) is πgr - closed in Z.
(gof)-1
(G) is πgr - closed in Z.
This shows that (gof)-1
: Y → Z is πgr - irresolute.
Hence (gof) is πgrc- homeomorphism.
Theorem :3.18
Let (Y, ) be πgr-T1/2- space. If f : X → Y and g : Y → Z are πgr- homeomorphism, then g f is a πgr-
homeomorphism.
Proof:
If f : X → Y and g : Y → Z be two πgr- homeomorphism. Let U be an open set in (X, τ). Since f is πgr- open
map, f(U) is πgr- open in Y.
Since Y is a πgr-T1/2- space, f(U) is regular open in Y and hence open in Y.
Also, since g is πgr- open map, g(f(U)) is πgr- open in Z.
6. On Πgr - Homeomorphisms In…
www.ijmsi.org 16 | P a g e
Hence (gof) (U) = g([f(U)] is πgr- open in Z for every open set U of X.
(gf) is a πgr- open map.
Let U be a closed set in Z .
Since g is πgr- continuous, g-1
(U) is πgr- closed in Y.
Since Y is a πgr-T1/2- space, every πgr- closed set in Y is regular closed in Y and hence closed in Y.
g-1
(V) is regular closed in Y and hence closed in Y.
Since f is πgr - continuous, f-1
[g-1
(V)] is πgr- closed set in X
(gof)-1
(V) is πgr-closed in X for every closed set V in Z.
(gof) is πgr-continuous and hence (gof) is a πgr-homeomorphism.
Remark: 3.19
Even though πgr- homeomorphism and πgrc- homeomorphism are independent concepts, we have the
following results( theorem 3.20 and theorem 3.21)
Theorem:3.20
Every πgr- homeomorphism from a πgr-T1/2- space into another πgr-T1/2- space is a homeomorphism.
Proof :
let f : X → Y be a πgr - homeomorphism. Then f is bijective, πgr- open and πgr- continuous map. Let U be an
open set in (X, τ).Since f is πgr- open and Y is πgr-T1/2- space, f(U) is πgr- open in Y.Since Y is a πgr-T1/2-
space, every πgr-open set is regular open in Y
f(U) is Regular open and hence open in Y.
f is an open map.
Let Y be a closed set in (Y, ). Since f is πgr- continuous, f-1
(V) is πgr-closed in X. Since X is a πgr-T1/2 -space,
every πgr - closed set is regular closed and hence closed in X.
Therefore, f is continuous.
Hence f is a homeomorphism.
Theorem:3.21
Every πgr- homeomorphism from a πgr-T1/2- space into another πgr-T1/2- space is a πgrc- homeomorphism.
Proof :
Let f : X → Y be a πgr - homeomorphism
Let U be πgr- closed in Y. Since Y is a πgr-T1/2- space, every πgr-closed set is regular closed and hence closed
in Y.
U is closed in Y.
Since f is πgr- continuous, f-1
(U) is πgr- closed in X.
Hence f is a πgr- irresolute map.
Let U be πgr- open set in X.
Since X is a πgr-T1/2- space, U is Regular open and hence open in X.
Since f is a πgr- open map, f(U) is πgr- open set in Y.
(f-1
)-1
= f ie, (f-1
)-1
(U) = f(U) is πgr- open in Y
Hence inverse image of (f-1
) is πgr- open in Y for every πgr- open set U of X and hence f-1
is πgr- irresolute.
Hence f is πgrc- homeomorphism.
Remark :3.22
Here, we shall introduce the group structure of the set of all πgrc- homeomorphism from a topological space
(X, τ) onto itself and denote it by πgrch-(X,τ).
Theorem :3.23
The set πgrch-(X,τ) is a group under composition of mappings.
Proof :
7. On Πgr - Homeomorphisms In…
www.ijmsi.org 17 | P a g e
We know that the composition of two πgrch(X,τ) is again a πgrch(X,τ).ie, For all f, g πgrch(X,τ), gf
πgrch(X, τ).We know that the composition of mappings is associative, the identity map belongs to πgrch(X,τ)
acts as an identity element. If f πgrch(X,τ),then f-1
πgrch(X, τ) such that f f-1
= f-1
f = I and so inverse
exists for each element of πgrch(X, τ).
Hence πgrc- homeomorphism (X, τ) is a group under the composition of mappings.
Theorem :3.24
Let f : (X, τ) →(Y, ) be a πgrc- homeomorphism. Then f induces an isomorphism from the group πgrch(X,τ)
onto the group πgrch(Y, ).
Proof :
We define a map,f* : πgrch(X,τ) → πgrch(Y, ) by f*(k) = f k f-1
every k πgrch(X,τ)
Then f* is a bijection and also for all k1, k2 πgrc- homeomorphism (X, τ)
f* (k1 k2) = f (k1 k2) f-1
= (fk1 f-1
) (fk2 f-1
)
= f* (k1) o f* (k2)
Hence f* is a homeomorphism and so it is an isomorphism induced by f.
Theorem :3.25
πgrc-homeomorphism is an equivalence relation in the collection of all topological spaces.
Proof :
Reflexivity and symmetry are immediate and transitivity follows from the fact that the composition of πgr-
irresolute maps is πgr-irresolute.
Proposition :3.26
For any two subsets A and B of (X, τ)
[1] If A B, then πgr- cl (A) πgr- cl (B)
[2] πgr- cl (AB) πgr- cl (A) πgr- cl (B)
Theorem :3.27
If f : (X, τ) → (Y, ) is a πgrc- homeomorphism and suppose πgr-closed set of X is closed under arbitrary
intersections, then πgr-cl(f-1
(B)) = f-1
(πgr- cl(B) for all B Y.
Proof :
Since f is a πgrc-homeomorphism, f and f-1
are πgr - irresolute.
Since f is π-irresolute, πgr - cl (f(B)) is a πgr - closed set in (Y, ), f-1
[πgr-cl (f(B)] is πgr - closed in (X, τ).
Now, f-1
(B) f-1
(πgr - cl f(B))
and πgr - cl (f-1
(B)) f-1
(πgr - cl(B))
Again, since f is a πgrc- homeomorphism, f-1
is πgr -irresolute. Since πgr - cl (f-1
(B)) is πgr - closed in X,
(f-1
)-1
[πgr - cl (f-1
(B))]= f (πgr - cl (f-1
(B)) is πgr - closed in Y.
Now, B (f-1
)-1
(f-1
(B))
(f-1
)-1
(πgr - cl (f-1
(B))
= f (πgr - cl (f-1
(B))
So, πgr - cl (B) f (πgr - cl (f-1
(B))
f-1
(πgr -cl(B)) πgr - cl (f-1
(B))
From & , the equality πgr-cl(f-1
(B)) = f-1
(πgr- cl(B) holds and hence the proof.
Corollary :3.28
If f : X → Y is a πgrc-homeomorphism, then πgr - cl (f(B)) =f(πgr - cl (B)) for all B X.
Proof :
Since f : X → Y is a πgrc- homeomorphism, f-1
: Y→ X is a πgrc- homeomorphism.
By previous theorem,
πgr - cl ((f-1
)-1
(B)) = (f-1
)-1
(πgr -cl(B)) for all B X
πgr -cl (f(B)) = f(πgr -cl(B))
8. On Πgr - Homeomorphisms In…
www.ijmsi.org 18 | P a g e
Corollary :3.29
If f : X →Y is a πgrc - homeomorphism, then f(πgr - int (B)) = πgr -int(f(B)) for all B X
Proof :
For any set B X, πgr -int (B) = [πgr - cl(BC
)]C
By previous corollary, we obtain
f (πgr -int (B)) = f [πgr - cl(BC
)C
]
= [f (πgr - cl(BC
)]C
= [πgr -cl(f(BC
)]C
= [πgr - cl(f(BC
)]C
= πgr - int(f(B))
Corollary :3.30
If f : X → Y is a πgrc- homeomorphism, then f-1
(πgr -int (B)) = πgr -int(f-1
(B)) for all B Y
Proof :
If f-1
: Y → X is also a πgrc - homeomorphism, the proof follows by using corollary 3.29.
REFERENCES
[1] I.Arokiarani, “Studies on generalizations of generalized closed sets and maps in topological spaces, Ph.D, Thesis, Bharathiar
University, Coimbatore(1997).
[2] D.A.Carnahan,”Some properties related to compactness in topological spaces ”, Ph.D. Thesis ,Univ.of Arkansas(1973).
[3] R.Devi, K.Balachandran H.Maki, “Semi generalized homoeomorphism and generalized semi-homeomorphism in topological
spaces, 26(1995),no .3,271-284.
[4] Dontchev.J, Noiri.T, “Quasi normal spaces and πg-closed sets”, Acta Math. Hungar , 89 (3), 2000,211-219.
[5] R. C. Jain, The role of regularly open sets in general topology, Ph. D. thesis, Meerut University, Institute of Advanced Studies,
Meerut, India 1980.
[6] C.Janaki and V.Jeyanthi,“On gr-separation axioms”, IJMER,Vol 4, (4), April -2014,7-13.
[7] V.Jeyanthi and C.Janaki, “gr-closed sets in topological spaces “,Asian Journal of current Engg. And Maths 1:5 , sep 2012, 241-
246.
[8] V.Jeyanthi. and Janaki ,” On πgr-continuous functions in topological spaces ”, IJERA , Vol 1, issue 3, Jan-Feb-2013.
[9] N. Levine, Generalized closed sets in topology, Rend. Cir. Mat. Palermo, 19(1970), 89- 96.
[10] N.Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70(1963), 36-41.
[11] P.E.Long and L.L.Herington, Basic properties of regular closed functions , Rend, Cir. Mat. Palermo, 27(1978),20-28.
[12] H.Maki,P.Sundaram, K.Balachandran, On Generalized homeomorphisms in topological spaces, Bull. Fukuoka Univ.Ed.Part-
III,40(1991),13-21.
[13] S.R.Malghan, Generalized closed maps, J.Karnatk Univ. Eci., 27(1982), 82-88.
[14] N.Nagaveni, Studies on generalizations of homeomorphisms in topological spaces, Ph.D, Thesis, Bharathiar University, Coimbatore
(1999).
[15] T.Noiri,”Mildly normal space and some functions”, Kyungpook Math .J.36(1996)183- 190.
[16] N.Palaniappan and K.C.Rao,”Regular generalized closed sets ” , Kyungpook Math .J. 33(1993), 211-219.
[17] M.K.Singal and A.R. Mathur, Anote on mildly compact spaces, Kyungpook. Math.J.9(1979),165-168.
[18] V.Zaitsav, On certain classes of topological spaces and their bicompactifications, Dokl Akad Nauk , SSSR (178), 778-779.