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 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.
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.
1) The document constructs expressions for vector fields Z that leave a p-form field Gp invariant in spacetime. This guarantees conservation of the integral of Gp over moving submanifolds.
2) Semi-explicit expressions are presented for Z in terms of Gp and auxiliary fields. They take different forms depending on properties of Gp, such as whether its exterior derivative is zero.
3) The expressions involve tensor equations that must be satisfied by the auxiliary fields. When written in coordinates, these become systems of partial differential equations whose solutions determine the auxiliary fields.
The document is a research paper that presents new results on odd harmonious graphs. It introduces the concepts of m-shadow graphs and m-splitting graphs. The paper proves that m-shadow graphs of paths and complete bipartite graphs are odd harmonious for all m ≥ 1. It also proves that n-splitting graphs of paths, stars and symmetric products of paths and null graphs are odd harmonious for all n ≥ 1. Additional families of graphs, including m-shadow graphs of stars and various graph constructions using paths, stars and their splitting graphs, are shown to admit odd harmonious labeling.
1. The document discusses the relationship between multicommodity flow problems and polyhedra related to Seymour's conjecture on binary clutters.
2. Seymour's conjecture states that a binary clutter is weakly bipartite if and only if it does not contain a forbidden minor like K5.
3. Weak bipartiteness of a signed graph is related to the cut condition being sufficient for the existence of multicommodity flows. If the signed graph is weakly bipartite, the cut condition is sufficient.
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.
1. The document discusses arithmetic progressions (AP) and geometric progressions (GP). An AP is a sequence where each term after the first is calculated by adding a constant to the previous term. A GP is a sequence where each term is calculated by multiplying the previous term by a constant.
2. Formulas are provided for calculating terms of APs and GPS, including formulas for the nth term, the sum of the first n terms, and identifying whether a set of numbers are in AP or GP.
3. The document concludes with 30 multiple choice questions testing understanding of APs and GPS.
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.
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.
1) The document constructs expressions for vector fields Z that leave a p-form field Gp invariant in spacetime. This guarantees conservation of the integral of Gp over moving submanifolds.
2) Semi-explicit expressions are presented for Z in terms of Gp and auxiliary fields. They take different forms depending on properties of Gp, such as whether its exterior derivative is zero.
3) The expressions involve tensor equations that must be satisfied by the auxiliary fields. When written in coordinates, these become systems of partial differential equations whose solutions determine the auxiliary fields.
The document is a research paper that presents new results on odd harmonious graphs. It introduces the concepts of m-shadow graphs and m-splitting graphs. The paper proves that m-shadow graphs of paths and complete bipartite graphs are odd harmonious for all m ≥ 1. It also proves that n-splitting graphs of paths, stars and symmetric products of paths and null graphs are odd harmonious for all n ≥ 1. Additional families of graphs, including m-shadow graphs of stars and various graph constructions using paths, stars and their splitting graphs, are shown to admit odd harmonious labeling.
1. The document discusses the relationship between multicommodity flow problems and polyhedra related to Seymour's conjecture on binary clutters.
2. Seymour's conjecture states that a binary clutter is weakly bipartite if and only if it does not contain a forbidden minor like K5.
3. Weak bipartiteness of a signed graph is related to the cut condition being sufficient for the existence of multicommodity flows. If the signed graph is weakly bipartite, the cut condition is sufficient.
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.
1. The document discusses arithmetic progressions (AP) and geometric progressions (GP). An AP is a sequence where each term after the first is calculated by adding a constant to the previous term. A GP is a sequence where each term is calculated by multiplying the previous term by a constant.
2. Formulas are provided for calculating terms of APs and GPS, including formulas for the nth term, the sum of the first n terms, and identifying whether a set of numbers are in AP or GP.
3. The document concludes with 30 multiple choice questions testing understanding of APs and GPS.
Stability criterion of periodic oscillations in a (15)Alexander Decker
This document presents the strong convergence of an iterative algorithm for solving the split common fixed point problem (SCFPP) in a real Hilbert space. It begins by introducing the SCFPP and related concepts like the split feasibility problem, common fixed point problem, and multiple set split feasibility problem. It then presents some preliminary definitions and lemmas regarding properties of operators and convergence. The main result is a theorem that proves the iterative sequence generated by a proposed algorithm converges strongly to a solution of the SCFPP, provided certain conditions on the operators are satisfied. This extends and improves on a previous result that only proved weak convergence.
Fractional integration and fractional differentiation of the product of m ser...Alexander Decker
This document presents theorems on fractional integrals and derivatives of the product of an M-series and H-function. An M-series is a special case of an H-function, and represents important functions in physics and applied sciences. Theorems are derived for the Riemann-Liouville fractional integral and derivative of the product. Additional theorems provide formulas for fractional integrals of the M-series alone, defined using an H-function operator previously established by Saxena and Khumbhat. The theorems extend previous work and provide new formulas incorporating these important special functions.
Some Fixed Point Theorems in b G -cone Metric Space Komal Goyal
The document introduces the concept of a bG-cone metric space, which generalizes both bG-metric spaces and cone metric spaces. Some key properties of bG-cone metric spaces are established, including uniqueness of limits, Cauchy sequences implying convergent sequences, and various characterizations of convergence. The paper then proves some fixed point theorems for maps satisfying general contractive conditions in the setting of bG-cone metric spaces. The results extend and include previous theorems from G-metric and cone metric spaces as special cases.
Some fixed point theorems of expansion mapping in g-metric spacesinventionjournals
Over the past two decades the development of fixed point theory in metric spaces has attracted
considerable attention due to numerous applications in areas such as variation and linear inequalities,
optimization and approximation theory. Therefore, different Authors proved many fixed points results for self
mapping defined on complete G-Metric space. The objectives of this study are to prove fixed point results for
mapping satisfying expansion conditions.
The document discusses using the renormalization group to build a "tower" of connected field theories across dimensions, starting from known 2-dimensional conformal field theories. It focuses on analyzing the O(N) x O(m) Landau-Ginzburg-Wilson model in 6 dimensions, which is connected to the 4D theory through a Wilson-Fisher fixed point. Perturbative calculations and the large N expansion are used to calculate critical exponents and check that the theories are in the same universality class. Studying higher dimensional theories could provide insight into physics beyond the Standard Model.
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.
Non-vacuum solutions of five dimensional Bianchi type-I spacetime in f (R) th...inventy
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.
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
This document proposes a generic method for representing and parsing various families of noncrossing digraphs using context-free grammars. It introduces a linear encoding that maps noncrossing digraphs to strings in a context-free language. This allows various constraints on digraph families to be expressed as star-free regular languages intersected with the encoding. This provides three representations for noncrossing digraphs and defines an ontology of digraph families. The method can be used for generic arc-factored parsing over any family by defining edge weights and using intersection with the family's constraints in a weighted context-free grammar.
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
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 presents an algorithm for calculating the number of spanning trees in chained graphs. It begins by reviewing relevant graph theory concepts like planar graphs, spanning trees, and recursive formulas for counting spanning trees using deletion/contraction and splitting methods. It then derives explicit recursions for counting spanning trees in families of graphs like wheel graphs, fan graphs, and corn graphs. The main result is a theorem providing a system of equations to calculate the number of spanning trees in a chained graph based on splitting it into components and accounting for the connecting paths. Applications to counting spanning trees in chained wheel graphs and chained corn graphs are discussed.
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 an earlier paper in 2017, Rastogi and Bajpai[1] defined and studied a special vector field of the first kind in a Finsler space as follows:
Definition 1: A vector field Xi(x), in a Finsler space, is said to be a special vector field of the first kind, if (i) Xi/j = - δij and (ii) Xi hij = Ɵj, where Ɵj is a non-zero vector field in the given Finsler space.
In 2019, some more special vector fields in a Finsler space of two and three dimensions have been defined and studied by the authors Dwivedi et al.[2] and Dwivedi et al.[3] In Dwivedi et al.[3], the authors defined and studied six kinds of special vector fields in a Finsler space of three dimensions and, respectively, called them special vector fields of the second, third, fourth, fifth, sixth, and seventh kind. In the present paper, we shall study some curvature properties of special vector fields of the first and seventh kind in a Finsler space of three dimensions.
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
Wild knots in higher dimensions as limit sets of kleinian groupsPaulo Coelho
This document summarizes a research paper that constructs wild knots in dimensions 3 through 7 as limit sets of Kleinian groups. The paper introduces the concepts of tangles and knots in higher dimensions, provides background on Kleinian groups, and describes a method of constructing orthogonal ball coverings of Euclidean spaces. Using this method, the paper constructs explicit orthogonal ball coverings for dimensions 1 through 5. It then shows how to construct knots as limit sets of Kleinian groups for these dimensions, and proves properties of the resulting knots, including that they are wildly embedded.
Strong convergence of an algorithm about strongly quasi nonexpansive mappingsAlexander Decker
This document presents an algorithm to solve the split common fixed-point problem (SCFPP) in Hilbert space. The algorithm is a modification of an existing algorithm for strongly quasi-nonexpansive operators. The author proves that under certain conditions, including the operators being demiclosed and the solution set being nonempty, the sequence generated by the algorithm converges strongly to a solution of the SCFPP. This extends and improves previous results on algorithms for solving split feasibility problems and common fixed-point problems.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal,
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.
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
Stability criterion of periodic oscillations in a (15)Alexander Decker
This document presents the strong convergence of an iterative algorithm for solving the split common fixed point problem (SCFPP) in a real Hilbert space. It begins by introducing the SCFPP and related concepts like the split feasibility problem, common fixed point problem, and multiple set split feasibility problem. It then presents some preliminary definitions and lemmas regarding properties of operators and convergence. The main result is a theorem that proves the iterative sequence generated by a proposed algorithm converges strongly to a solution of the SCFPP, provided certain conditions on the operators are satisfied. This extends and improves on a previous result that only proved weak convergence.
Fractional integration and fractional differentiation of the product of m ser...Alexander Decker
This document presents theorems on fractional integrals and derivatives of the product of an M-series and H-function. An M-series is a special case of an H-function, and represents important functions in physics and applied sciences. Theorems are derived for the Riemann-Liouville fractional integral and derivative of the product. Additional theorems provide formulas for fractional integrals of the M-series alone, defined using an H-function operator previously established by Saxena and Khumbhat. The theorems extend previous work and provide new formulas incorporating these important special functions.
Some Fixed Point Theorems in b G -cone Metric Space Komal Goyal
The document introduces the concept of a bG-cone metric space, which generalizes both bG-metric spaces and cone metric spaces. Some key properties of bG-cone metric spaces are established, including uniqueness of limits, Cauchy sequences implying convergent sequences, and various characterizations of convergence. The paper then proves some fixed point theorems for maps satisfying general contractive conditions in the setting of bG-cone metric spaces. The results extend and include previous theorems from G-metric and cone metric spaces as special cases.
Some fixed point theorems of expansion mapping in g-metric spacesinventionjournals
Over the past two decades the development of fixed point theory in metric spaces has attracted
considerable attention due to numerous applications in areas such as variation and linear inequalities,
optimization and approximation theory. Therefore, different Authors proved many fixed points results for self
mapping defined on complete G-Metric space. The objectives of this study are to prove fixed point results for
mapping satisfying expansion conditions.
The document discusses using the renormalization group to build a "tower" of connected field theories across dimensions, starting from known 2-dimensional conformal field theories. It focuses on analyzing the O(N) x O(m) Landau-Ginzburg-Wilson model in 6 dimensions, which is connected to the 4D theory through a Wilson-Fisher fixed point. Perturbative calculations and the large N expansion are used to calculate critical exponents and check that the theories are in the same universality class. Studying higher dimensional theories could provide insight into physics beyond the Standard Model.
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.
Non-vacuum solutions of five dimensional Bianchi type-I spacetime in f (R) th...inventy
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.
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
This document proposes a generic method for representing and parsing various families of noncrossing digraphs using context-free grammars. It introduces a linear encoding that maps noncrossing digraphs to strings in a context-free language. This allows various constraints on digraph families to be expressed as star-free regular languages intersected with the encoding. This provides three representations for noncrossing digraphs and defines an ontology of digraph families. The method can be used for generic arc-factored parsing over any family by defining edge weights and using intersection with the family's constraints in a weighted context-free grammar.
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
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 presents an algorithm for calculating the number of spanning trees in chained graphs. It begins by reviewing relevant graph theory concepts like planar graphs, spanning trees, and recursive formulas for counting spanning trees using deletion/contraction and splitting methods. It then derives explicit recursions for counting spanning trees in families of graphs like wheel graphs, fan graphs, and corn graphs. The main result is a theorem providing a system of equations to calculate the number of spanning trees in a chained graph based on splitting it into components and accounting for the connecting paths. Applications to counting spanning trees in chained wheel graphs and chained corn graphs are discussed.
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 an earlier paper in 2017, Rastogi and Bajpai[1] defined and studied a special vector field of the first kind in a Finsler space as follows:
Definition 1: A vector field Xi(x), in a Finsler space, is said to be a special vector field of the first kind, if (i) Xi/j = - δij and (ii) Xi hij = Ɵj, where Ɵj is a non-zero vector field in the given Finsler space.
In 2019, some more special vector fields in a Finsler space of two and three dimensions have been defined and studied by the authors Dwivedi et al.[2] and Dwivedi et al.[3] In Dwivedi et al.[3], the authors defined and studied six kinds of special vector fields in a Finsler space of three dimensions and, respectively, called them special vector fields of the second, third, fourth, fifth, sixth, and seventh kind. In the present paper, we shall study some curvature properties of special vector fields of the first and seventh kind in a Finsler space of three dimensions.
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
Wild knots in higher dimensions as limit sets of kleinian groupsPaulo Coelho
This document summarizes a research paper that constructs wild knots in dimensions 3 through 7 as limit sets of Kleinian groups. The paper introduces the concepts of tangles and knots in higher dimensions, provides background on Kleinian groups, and describes a method of constructing orthogonal ball coverings of Euclidean spaces. Using this method, the paper constructs explicit orthogonal ball coverings for dimensions 1 through 5. It then shows how to construct knots as limit sets of Kleinian groups for these dimensions, and proves properties of the resulting knots, including that they are wildly embedded.
Strong convergence of an algorithm about strongly quasi nonexpansive mappingsAlexander Decker
This document presents an algorithm to solve the split common fixed-point problem (SCFPP) in Hilbert space. The algorithm is a modification of an existing algorithm for strongly quasi-nonexpansive operators. The author proves that under certain conditions, including the operators being demiclosed and the solution set being nonempty, the sequence generated by the algorithm converges strongly to a solution of the SCFPP. This extends and improves previous results on algorithms for solving split feasibility problems and common fixed-point problems.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal,
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.
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
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
This document introduces and studies a new type of closed set called strongly τb-closed (τ*b-closed) sets. The following is summarized:
1. τ*b-closed sets are between closed sets and gsg-closed sets.
2. Properties and relationships of τ*b-closed sets are investigated, showing they are finer than τb-closed sets and contained within several other closed set classes.
3. Characterizations of τ*b-closed and τ*b-open sets are provided, such as the union of τ*b-closed sets being τ*b-closed.
γ Regular-open sets and γ-extremally disconnected spacesAlexander Decker
This document introduces γ-regular-open sets in topological spaces with an operation γ. γ-Regular-open sets lie between γ-clopen and γ-open sets. The complement of a γ-regular-open set is γ-regular-closed. Several properties of γ-regular-open and γ-regular-closed sets are proved, including: the intersection of two γ-regular-open sets is γ-regular-open, and the union of two γ-regular-closed sets is γ-regular-closed. The concepts of γ-regular-open and regular open sets are shown to be independent, but coincide for γ-regular spaces.
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.
IRJET- W-R0 Space in Minimal G-Closed Sets of Type1IRJET Journal
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On Decomposition of gr* - closed set in Topological Spaces
1. International Journal Of Mathematics And Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.Ijmsi.Org || Volume 2 Issue 11|| December. 2014 || PP-45-55
www.ijmsi.org 45 | P a g e
On Decomposition of gr*
- closed set in Topological Spaces
1
K.Indirani, 2
P.Sathishmohan and 3
, V.Rajendran
1
Department of Mathematics, Nirmala College for Woman, Coimbatore, TN, India.
2,3
Department of Mathematics, KSG college, Coimbatore, TN, India.
ABSTRACT: The aim of this paper is to introduced and study the classes of gr* locally closed set and
different notions of generalization of continuous functions namely gr*lc-continuity, gr*lc**-continuity and
gr*lc**-continuity and their corresponding irresoluteness were studied. Furthermore, the notions of Z-sets, Zr-
sets and Zr*-sets are used to obtain decompositions of gr*-continuity, gr*-open maps and contra gr*-continuity
were investigated.
KEYWORDS: gr*-separated, gr*
-dense, gr*
-submaximal, gr*lc-continuity, gr*lc**-continuity gr*lc**-
continuity, contra Z- continuity, contra Zr- continuity and contra Zr*- continuity
I. INTRODUCTION:
The first step of locally closedness was done by Bourbaki [2]. He defined a set A to be locally closed if
it is the intersection of an open and a closed set. In literature many general topologists introduced the studies of
locally closed sets. Extensive research on locally closedness and generalizing locally closedness were done in
recent years. Stone [10] used the term LC for a locally closed set. Ganster and Reilly used locally closed sets in
[4] to define LC-continuity and LC-irresoluteness. Balachandran et al [1] introduced the concept of generalized
locally closed sets. The aim of this paper is to introduce and study the classes of gr* locally closed set and
different notions of generalization of continuous functions namely gr*lc-continuity, gr*lc**-continuity and
gr*lc**-continuity and their corresponding irresoluteness were studied. Furthermore, the notions of Z-sets, Zr-
sets and Zr*-sets are used to obtain decompositions of gr*-continuity, gr*-open maps and contra gr*-continuity
were investigated.
II. PRELIMINARY NOTES
Throughout this paper (X,τ), (Y,σ) are topological spaces with no separation axioms assumed unless
otherwise stated. Let AX. The closure of A and the interior of A will be denoted by Cl(A) and Int(A)
respectively.
Definition 2.1. A Subset S of a space (X,τ) is called
(i) locally closed (briefly lc )[4] if S=UF, where U is open and F is closed in (X,τ).
(ii) r-locally closed (briefly rlc ) if S=UF, where U is r-open and F is r-closed in (X,τ).
(iii) generalized locally closed (briefly glc ) [1] if S=UF, where U is g-open and F is g-closed in (X,τ).
Definition 2.2. [8] For any subset A of (X,τ), RCl(A)=∩{G: GA, G is a regular closed subset of X}.
Definition 2.3. [5] A subset A of a topological space (X,τ) is called a generalized regular star closed set [
briefly gr*
-closed ] if Rcl(A)⊆U whenever A⊆U and U is g-open subset of X.
Definition 2.4. [6] For a subset A of a space X, gr*-cl(A) = ⋂{F: A⊆F, F is gr* closed in X} is called the gr*-
closure of A.
Remark 2.5. [5] Every r-closed set in X is gr*
-closed in X.
Remark 2.6. For a topological space (X,τ), the following statements hold:
(1) Every closed set is gr*-closed but not conversely [5].
(2) Every gr*-closed set is g-closed but not conversely [5].
(3) Every gr*-closed set is sg-closed but not conversely [5].
(4) A subset A of X is gr*-colsed if and only if gr*-cl(A)=A.
(5) A subset A of X is gr*-open if and only if gr*-int(A)=A.
Corollary 2.7. If A is a gr*-closed set and F is a closed set, then A∩F is a gr*-closed set.
Definition 2.8. A function f:(X,τ)→(Y,σ) is called
2. On Decomposition of gr*
- closed set in Topological Spaces
www.ijmsi.org 46 | P a g e
i) LC-continuous [4] if f-1
(V)∈ LC(X,τ) for every V∈σ.
ii) GLC-continuous [1] if f-1
(V)∈ GLC(X,τ) for every V∈σ.
Definition 2.9. A subset S of a space (X,τ) is called
(i) submaximal [3] if every dense subset is open.
(ii) g-submaximal [1] if every dense subset is g-open.
(iii) rg-submaximal [2] if every dense subset is rg-open.
Definition 2.10. A subset A of a space (X,τ) is called
(i) an *-set [7] if int(A) = int(cl(int(A)).
(ii) an A-set [11] if A = GF where G is open and F is regular closed in X.
(iii) a t-set [12] if int (A) = int(cl(A)).
(iv) a C-set [9] if A = GF where G is open and F is a t-set in X.
(v) a Cr-set [9] if A = GF where G is rg-open and F is a t-set in X.
(vi) a Cr*-set [9] if A = GF where G is rg-open and F is an *-set in X.
III. GR*
LOCALLY CLOSED SET
Definition 3.1. A subset A of (X,τ) is said to be generalized regular star locally closed set (briefly gr*
lc) if
S=L∩M where L is gr*
-open and M is gr*
-closed in (X,τ).
Definition 3.2. A subset A of (X,τ) is said to be gr*
lc*
set if there exists a gr*
-open set L and a closed set M of
(X,τ) such that S=L∩M.
Definition 3.3. A subset B of (X,τ) is said to be gr*
lc**
set if there exists an open set L and a gr*
-closed set M
such that B=L∩M.
The class of all gr*
lc (resp. gr*
lc*
& gr*
lc**
) sets in X is denoted by GR*
LC(X).(resp. GR*
LC*
(X) &
GR*
LC**
(X))
From the above definitions we have the following results.
Proposition 3.4.
i) Every locally closed set is gr*
lc.
ii) Every rlc-set is gr*
lc.
iii) Every gr*
lc*
-set is gr*
lc.
iv) Every gr*
lc**
-set is gr*
lc.
v) Every gr*
lc-set is glc.
vi) Every rlc-set is gr*
lc*
.
vii) Every rlc-set is gr*
lc**
.
viii) Every rlc*
-set is gr*
lc**
.
ix) Every rlc**
-set is gr*
lc*
.
x) Every rlc**
-set is gr*
lc.
xi) Every gr*
lc*
-set is glc.
xii) Every gr*
-closed set is gr*
lc.
However the converses of the above are not true as seen by the following examples
Example 3.5. Let X={a,b,c,d} with τ={ϕ,{c},{a,b},{a,b,c},X}. Then A={a} is gr*
lc-set but not locally closed.
Example 3.6. Let X={a,b,c,d} with τ={ϕ,{a},{b},{a,b},{a,b,c},X}. Then A={c} is gr*
lc-set but not rlc-set.
Example 3.7. In example 3.5, Let A={d} is gr*
lc-set but not gr*
lc*
-set.
Example 3.8. In example 3.6, Let A={c} is gr*
lc-set but not gr*
lc**
-set.
Example 3.9. Let Let X={a,b,c,d} with τ={ϕ,{a,d},{b,c},X}. Then A={a} is glc-set but not gr*
lc-set.
Example 3.10. In Example 3.5, Let A={b} is gr*
lc*
set but not rlc-set.
Example 3.11. In example 3.6, Let A={d} is gr*
lc**
set but not rlc-set.
Example 3.12. Let X={a,b,c,d} with τ={ϕ,{b},{c},{b,c},{b,c,d},X}. Then A={a,b} is gr*
lc**
-set but not rlc*
-set.
3. On Decomposition of gr*
- closed set in Topological Spaces
www.ijmsi.org 47 | P a g e
Example 3.13. In example 3.12, Let A={d} is gr*
lc*
set but not rlc**
-set.
Example 3.14. In example 3.12, Let A={b} is gr*
lc-set but not rlc**
-set.
Example 3.15. In example 3.6, Let A={d} is glc-set but not gr*
lc*
-set.
Example 3.16. Let X={a,b,c,d} with τ={ϕ,{a},{c},{a,c},{a,c,d},X}. Then A={a} is gr*
lc-set but not gr*
-closed
set.
Remark 3.17.The concepts of gr*
lc*
set and gr*
lc**
sets are independent of each other as seen from the following
example.
Example 3.18. In example 3.6, Let A={c} is gr*
lc*
-set but not gr*
lc**
-set and Let A={d} is gr*
lc**
-set but not
gr*
lc*
-set.
Remark 3.19. The concepts of gr*
lc*
-set and rlc*
-set are independent of each other as seen form the following
example.
Example 3.20. Let X={a,b,c,d} with τ={ϕ,{a},{c},{a,c},{a,c,d},X}. Then A={d} is gr*
lc*
-set but not rlc*
-set
and let A={b} is rlc*
-set but not gr*
lc*
-set.
Remark 3.21. The concepts of gr*
lc**
-set and rlc**
-set are independent of each other as seen from the following
example.
Example 3.22. In example 3.20, Let A={b} is gr*
lc**
-set but not rlc**
-set and let A={a,c} is rlc**
-set but not
gr*
lc**
-set.
Remark 3.23. The concepts of locally closed set and gr*
lc*
are independent of each other as seen from the
following example.
Example 3.24. In example 3.20, Let A={b,d} is locally closed set but not gr*
lc*
-set and let A={d} is gr*
lc*
-set
but not locally closed set.
Remark 3.25. The concepts of rlc**
set and gr*
closed-set are independent of each other as seen from the
following example.
Example 3.26. In example 3.5, Let A={c} is rlc**
set but not gr*
closed set and let A={d} is gr*
-closed set but
not rlc**
-set.
Remark 3.27. Union of two gr*
lc-sets are gr*
lc-sets.
Remark 3.28. Union of two gr*
lc*
-sets(resp. gr*
lc**
-set) need not be an gr*
lc*
-set(resp. gr*
lc**
-set) as seen from
the following examples.
Example 3.29. In example 3.5, Then the sets {a} and {c,d} are gr*
lc*
-sets, but their union {a,c,d}∉ gr*
lc*
(X).
Example 3.30. In example 3.6, then the sets {a} and {b} are gr*
lc**
-sets but their union {a,b}∉ gr*
lc**
(X).
The above discussions are summarized in the following implications..
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IV. GR*
-DENSE SETS AND GR*
-SUBMAXIMAL SPACES
Definition 4.1. A subset A of (X,τ) is called gr*
-dense if gr*
-cl(A)=X.
Example 4.2. Let X={a,b,c,d}with τ={ϕ,{b},{a,c},{a,b,c},X}. Then the set A={a,b,c} is gr*
-dense in (X,τ).
Recall that a subset A of a space (X,τ) is called dense if cl(A)=X.
Proposition 4.3. Every gr*
-dense set is dense.
Let A be an gr*
-dense set in (X,τ). Then gr*
-cl(A)=X. Since gr*
-cl(A)⊆rcl(A)⊆cl(A). we have cl(A)=X and so A
is dense.
The converse of the above proposition need not be true as seen from the following example.
Example 4.4. Let X={a,b,c,d}with τ={ϕ,{b},{c,d},{b,c,d},X}. Then the set A={b,c} is a dense in (X,τ) but it is
not gr*
-dense in (X,τ).
Definition 4.5. A topological space (X,τ) is called gr*
-submaximal if every dense subset in it is gr*
-open in
(X,τ).
Proposition 4.6. Every submaximal space is gr*
-submaximal.
Proof. Let (X,τ) be a submaximal space and A be a dense subset of (X,τ). Then A is open. But every open set is
gr*
-open and so A is gr*
-open. Therefore (X,τ) is gr*
-submaximal.
The converse of the above proposition need not be true as seen from the following example.
Example 4.7. Let X = {a,b,c}with τ = {ϕ,X}. Then gr*
O(X) = P(X). we have every dense subset is gr*
-open and
hence (X,τ) is gr*
-submaximal. However, the set A={c} is dense in (X,τ), but it is not open in (X,τ). Therefore
(X,τ) is not submaximal.
Proposition 4.8. Every gr*
-submaximal space is g-submaximal.
Proof. Let (X,τ) be a gr*
-submaximal space and A be a dense subset of (X,τ). Then A is gr*
-open. But every gr*
-
open set is g-open and A is g-open. Therefore (X,τ) is g-submaximal.
The converse of the above proposition need not be true as seen from the following example.
Example 4.9. Let X = {a,b,c,d}with τ = {ϕ,{d},{a,b,c},X}. Then GO(X) = P(X) and
gr*
O(X)={ϕ,{d},{a,b,c},X}. we have every dense subset is g-open and hence (X,τ) is g-submaximal. However,
the set A={a} is dense in (X,τ), but it is not gr*
-open in (X,τ). Therefore (X,τ) is not gr*
-submaximal.
Proposition 4.10. Every gr*
-submaximal space is rg-submaximal.
The converse of the above proposition need not be true as seen from the following example.
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Example 4.11. Let X={a,b,c,d}with τ={ϕ,{a,d},{b,c},X}. Then RO(X)=P(X) and
gr*
O(X)={ϕ,{b,c},{a,d},X}.Every dense subset is rg-open and hence (X,τ) is rg-submaximal. However the set
A={a} is dense in (X,τ), but it is not gr*
-open in (X,τ). Therefore (X,τ) in not gr*
-submaximal.
Remark 4.12. From propositions 4.6, 4.8 and 4.10, we have the following diagram.
Submaximal gr*
-submaximal g-submaximal rg-maximal
Theorem 4.13. Assume that GR*
C(X) is closed under finite intersections. For a subset A of (X,τ) the following
statements are equivalent:
(1) A∈ GR*
LC(X),
(2) A=S∩gr*-cl(A) for some gr*-open set S,
(3) gr*-cl(A)−A is gr*-closed,
(4) A∪(gr*-cl(A))c
is gr*-open,
(5) A⊆ gr*-int(A∪( gr*-cl(A))c
).
Proof. (1)⇒(2). Let A∈GR*LC(X). Then A=S∩G where S is gr*-open and G is gr*-closed. Since A⊆G, gr*-
cl(A)⊆G and so S∩ gr*-cl(A)⊆A. Also A⊆S and A⊆ gr*-cl(A) implies A⊆S∩ gr*-cl(A) and therefore A=S∩
gr*-cl(A).
(2)⇒(3). A=S∩ gr*-cl(A) implies gr*-cl(A)−A= gr*-cl(A)∩Sc
which is gr*-closed since Sc
is gr*-closed and
gr*-cl(A) is gr*-closed.
(3)⇒(4). A∪( gr*-cl(A))c
=( gr*-cl(A)−A)c
and by assumption, (gr*-cl(A)−A)c
is gr*-open and so is A∪( gr*-
cl(A))c
.
(4)⇒(5). By assumption, A∪( gr*-cl(A))c
= gr*-int(A∪( gr*-cl(A))c
) and hence A⊆ gr*-int(A∪( gr*-cl(A))c
).
(5)⇒(1). By assumption and since A⊆gr*-cl(A), A=gr*-int(A∪(gr*-cl(A))c
)∩gr*-cl(A). Therefore,
A∈GR*LC(X).
Theorem 4.14. For a subset A of (X,τ), the following statements are equivalent:
(1) A∈GR*LC*
(X),
(2) A=S∩cl(A) for some gr*-open set S,
(3) cl(A)−A is gr*-closed,
(4) A∪(cl(A))c
is gr*-open.
Proof. (1)⇒(2). Let A∈GR*LC*
(X). There exist an gr*-open set S and a colsed set G such that A=S∩G. Since
A⊆S and A⊆cl(A), A⊆S∩cl(A). Also since cl(A)⊆G, S∩cl(A)⊆S∩G=A. Therefore A=S∩cl(A).
(2)⇒(1). Since S is gr*-open and cl(A) is a closed set, A=S∩cl(A)∈GR*LC*
(X).
(2)⇒(3). Since cl(A)−A=cl(A)∩Sc
, cl(A)−A is gr*-closed by Remark 2.6.
(3)⇒(2). Let S=(cl(A)−A)c
. Then by assumption S is gr*-open in (X,τ) and A=S∩cl(A).
(3)⇒(4). Let G=cl(A)−A. Then Gc
=A∪(cl(A))c
and A∪(cl(A))c
is gr*-open.
(4)⇒(3). Let S=A∪(cl(A))c
. Then Sc
is gr*-closed and Sc
=cl(A)−A and so cl(A)−A is gr*-closed.
Theorem 4.15. A space (X,τ) is gr*-submaximal if and only if P(X)=GR*LC*
(X).
Proof. Necessity. Let A∈P(X) and let V=A∪(cl(A))c
. This implies that cl(V)=cl(A)∪(cl(A))c
=X. Hence
cl(V)=X. Therefore V is a dense subset of X. Since (X,τ) is gr*-submaximal, V is gr*-open. Thus A∪(cl(A))c
is
gr*-open and by theorem 4.14, we have A∈GR*LC*
(X).
Sufficiency. Let A be a dense subset of (X,τ). This implies A∪(cl(A))c
=A∪Xc
=A∪ϕ=A. Now A∈GR*LC*
(X)
implies that A=A∪(cl(A))c
is gr*-open by Theorem 4.14. Hence (X,τ) is gr*-submaximal.
Theorem 4.16. Let A be a subset of (X,τ). Then A∈GR*LC**
(X) if and only if A=S∩gr*-cl(A) for some open
set S.
Proof. Let A∈GR*LC**
(X). Then A=S∩G where S is open and G is gr*-closed. Since A⊆G, gr*-cl(A)⊆G. We
obtain A=A∩gr*-cl(A)=S∩G∩gr*-cl(A)=S∩gr*-cl(A).
Converse part is trivial.
Theorem 4.17. Let A be a subset of (X,τ). If A∈GR*LC**
(X), then gr*-cl(A)−A is gr*-closed and A∪(gr*-
cl(A))c
is gr*-open.
Proof. Let A∈GR*LC**
(X). Then by theorem 4.16, A=S∩gr*-cl(A) for some open set S and gr*-cl(A)−A=gr*-
cl(A)∩Sc
is gr*-closed in (X,τ). If G=gr*-cl(A)−A, then Gc
=A∪(gr*-cl(A))c
and Gc
is gr*-open and so is
A∪(gr*-cl(A))c
.
Proposition 4.18. Assume that GR*O(X) forms a topology. For subsets A and B in (X,τ), the following are true:
(1) If A, B∈GR*LC(X), then A∩B∈GR*LC(X).
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(2) If A, B∈GR*LC*
(X), then A∩B∈GR*LC*
(X).
(3) If A, B∈GR*LC**
(X), then A∩B∈GR*LC**
(X).
(4) If A∈GR*LC(X) and B is gr*-open (resp. gr*-closed), then A∩B∈GR*LC(X).
(5) If A∈GR*LC*
(X) and B is gr*-open (resp. closed), then A∩B∈GR*LC*
(X).
(6) If A∈GR*LC**
(X) and B is gr*-closed (resp. open), then A∩B∈GR*LC**
(X).
(7) If A∈GR*LC*
(X) and B is gr*-closed, then A∩B∈GR*LC(X).
(8) If A∈GR*LC**
(X) and B is gr*-open, then A∩B∈GR*LC(X).
(9) If A∈GR*LC**
(X) and B∈GR*LC*
(X), then A∩B∈GR*LC(X).
Proof. By Remark 2.5 and Remark 2.6, (1) to (8) hold.
(9). Let A=S∩G where S is open and G is gr*-closed and B=P∩Q where P is gr*-open and Q is closed. Then
A∩B=( S∩P)∩(G∩Q) where S∩P is gr*-open and G∩Q is gr*-closed, by Remark 2.6. Therefore
A∩B∈GR*LC(X).
Definition 4.19 . Let A and B be subsets of (X,τ). Then A and B are said to be gr*-separated if A∩gr*-cl(B)=ϕ
and gr*-cl(A)∩B=ϕ.
Example 4.20. Let X={a,b,c}with τ={ϕ,{a},{c},{a,c},{b,c},X}. Let A={a} and B={b}. Then gr*
-cl(A)={a}
and gr*
-cl(B)={b} and so the sets A and B are gr*
-separated.
Proposition 4.21. Assume that GR*O(X) forms a topology. For a topological space (X,τ), the following are
true:
(1) Let A, B∈ GR*LC(X). If A and B are gr*-separated then A∪B∈ GR*LC(X).
(2) Let A, B∈ GR*LC*
(X). If A and B are separated (i.e., A∩cl(B)=ϕ and cl(A)∩B=ϕ), then A∪B∈
GR*LC*
(X).
(3) Let A, B∈ GR*LC**
(X). If A and B are gr*-separated then A∪B∈ GR*LC**
(X).
Proof. (1) Since A, B∈ GR*LC(X). by theorem 4.13, there exists gr*-open sets U and V of (X,τ) such that
A=U∩ gr*-cl(A) and B=V∩ gr*-cl(B). Now G=U∩(X− gr*-cl(B)) and H=V∩(X− gr*-cl(A)) are gr*-open
subsets of (X,τ). Since A∩ gr*-cl(B)=ϕ, A⊆( gr*-cl(B))c
. Now A=U∩ gr*-cl(A) becomes A∩( gr*-cl(B))c
=G∩
gr*-cl(A). Then A=G∩ gr*-cl(A). Similarly B=H∩ gr*-cl(B). Moreover G∩ gr*-cl(B)=ϕ and H∩ gr*-cl(A)=ϕ.
Since G and H are gr*-open sets of (X,τ), G∪H is gr*-open. Therefore A∪B=(G∪H)∩ gr*-cl(A∪B) and hence
A∪B∈ GR*LC(X).
(2) and (3) are similar to (1), using Theorems 4.13 and 4.14.
Lemma 4.22. If A is gr*
-closed in (X,τ) and B is gr*
-closed in (Y,σ), then A×B is gr*
-closed in (X×Y, τ×σ).
Theorem 4.23. Let (X,τ) and (Y,σ) be any two topological spaces. Then
i) If A∈GR*LC(X,τ) and B∈GR*LC(Y,σ), then A×B∈GR*LC(X×Y, τ×σ).
ii) If A∈GR*LC*
(X,τ) and B∈GR*LC*(Y,σ), then A×B∈GR*LC*(X×Y, τ×σ).
iii) If A∈GR*LC**
(X,τ) and B∈GR*LC**(Y,σ), then A×B∈GR*LC**(X×Y, τ×σ).
Proof. Let A∈GR*LC(X,τ) and B∈GR*LC(Y,σ). Then there exists gr*
-open sets V and V’
of (X,τ) and (Y,σ)
respectively and gr*
-closed sets W and W’
of (X,τ) and (Y,σ) respectively such that A=V∩W and B=V’
∩W’
.
Then A×B= (V∩W)×(V’
∩W’
)=(V×V’
)∩(W×W’
) holds and hence A×B∈GR*LC(X×Y, τ×σ).
The proofs of (ii) and (iii) are similar to (i).
V. GR*
LC -CONTINUOUS AND GR*
LC-IRRESOLUTE FUNCTIONS
In this section, we define gr*
LC-continuous and gr*
LC-irresolute functions and obtain a pasting lemma
for gr*
LC**-continuous functions and irresolute functions.
Definition 5.1. A function f:(X,τ)→(Y,σ) is called
i) gr*
LC-continuous if f-1
(V)∈ gr*
LC(X,τ) for every V∈σ.
ii) gr*
LC*
-continuous if f-1
(V)∈ gr*
LC*
(X,τ) for every V∈σ.
iii) gr*
LC**
-continuous if f-1
(V)∈ gr*
LC**
(X,τ) for every V∈σ.
iv) gr*
LC-irresolute if f-1
(V)∈gr*
LC(X,τ) for every V∈gr*
LC(Y,σ).
v) gr*
LC*
-irresolute if f-1
(V)∈gr*
LC*
(X,τ) for every V∈gr*
LC*(Y,σ).
vi) gr*
LC**
-irresolute if f-1
(V)∈gr*
LC**
(X,τ) for every V∈gr*
LC**(Y,σ).
Proposition 5.2. If f:(X,τ)→(Y,σ) is gr*
LC-irresolute, then it is gr*
LC-continuous.
Proof. Let V be open in Y. Then V∈gr*
LC(Y,σ). By assumption, f-1
(V)∈gr*
LC(X,τ). Hence f is gr*
LC-
continuous.
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Proposition 5.3. Let f:(X,τ)→(Y,σ) be a function, then
1. If f is LC-continuous, then f is gr*
LC-continuous.
2. If f is gr*
LC*
-continuous, then f is gr*
LC-continuous.
3. If f is gr*
LC**
-continuous, then f is gr*
LC-continuous.
4. If f is gr*
LC-continuous, then f is glc-continuous.
Remark 5.4. The converses of the above are not true may be seen by the following examples.
Example 5.5. 1. Let X=Y={a,b,c,d}, τ={ϕ,{c},{a,b},{a,b,c},X} and ={ϕ,{a},{a,d},X}. Let f: X→Y be the
identity map. Then f is gr*
LC-continuous but not LC-continuous. Since for the open set {a,d}, f-1
{a,d}= {a,d} is
not locally closed in X.
2. Let X=Y={a,b,c,d}, τ={ϕ,{c},{a,b},{a,b,c},X} and ={ϕ,{d},{a,d},{a,c,d},X}. Let f: X→Y be the identity
map. Then f is gr*
LC-continuous but not gr*
LC*
-continuous. Since for the open set {a,c,d}, f-1
{a,c,d} = {a,c,d}is
not gr*
LC*
-closed in X.
3. Let X=Y={a,b,c,d}, τ={ϕ,{c},{a,b},{a,b,c},X} and ={ϕ,{a},{a,c},X}. Let f: X→Y be the identity map.
Then f is gr*
LC-continuous but not gr*
LC**
-continuous. Since for the open set {a,c}, f-1
{a,c} = {a,c} is not
gr*
LC**
-closed in X.
4. Let X=Y={a,b,c,d}, τ={ϕ,{a,d},{b,c},X} and ={ϕ,{a},{a,b},{a,b,d},X}. Let f: X→Y be the identity map.
Then f is glc-continuous but not gr*
LC-continuous. Since for the open set {a,b,d}, f-1
{a,b,d} = {a,b,d} is not
gr*
LC-set in X.
We recall the definition of the combination of two funtions: Let X=A∪B and f:A→Y and h:B→Y be
two functions. We say that f and h are compatible if f∕A∩B=h∕A∩B. If f:A→Y and h:B→Y are compatible,
then the functions (f△h)(X)=h(X) for every x∈B is called the combination of f and h.
Pasting lemma for gr*
LC**
-continuous (resp. gr*
LC**
-irresolute) functions.
Theorem 5.6. Let X=A∪B, where A and B are gr*
-closed and regular open subsets of (X,τ) and f:(A,
τ∕B)→(Y,σ) and h:(B,τ∕B)→(Y,σ) be compatible functions.
a) If f and h are gr*
LC**
-continuous, then (f△h):X→Y is gr*
LC**
-continuous.
b) If f and h are gr*
LC**
-irresolute, then (f△h):X→Y is gr*
LC**
-irresolute.
Next we have the theorem concerning the composition of functions.
Theorem 5.7. Let f:(X,τ)→(Y,σ) and g:(Y,σ)→(Z,η) be two functions, then
a) g∘f is gr*
LC-irresolute if f and g are gr*
LC-irresolute.
b) g∘f is gr*
LC*
-irresolute if f and g are gr*
LC*
-irresolute.
c) g∘f is gr*
LC**
-irresolute if f and g are gr*
LC**
-irresolute.
d) g∘f is gr*
LC-continuous if f is gr*
LC-irresolute and g is gr*
LC-continuous.
e) g∘f is gr*
LC*
-continuous if f is gr*
LC*
-continuous and g is continuous.
f) g∘f is gr*
LC-continuous if f is gr*
LC-continuous and g is continuous.
g) g∘f is gr*
LC*
-continuous if f is gr*
LC*
-irresolute and g is gr*
LC*
-continuous.
h) g∘f is gr*
LC**
-continuous if f is gr*
LC**
-irresolute and g is gr*
LC**
-continuous.
VI. DECOMPOSITION OF GR*
-CLOSED SETS
In this section, we introduce the notions of Z-sets, Zr-sets and Zr*-sets to obtain decompostions of gr*
-
closed sets.
Definition 6.1. A subset S of (X,τ) is called a
1) Z-set if S=L∩M where L is gr*
-open and M is a t-set.
2) Zr-set if S=L∩M where L is gr*
-open and M is a α*
-set.
3) Zr*-set if S=L∩M where L is gr*
-open and M is a A-set.
Proposition 6.2.
1. Every Z-set is a C-set.
2. Every Z-set is a Cr-set.
3. Every Z-set is a Cr*-set.
4. Every Zr-set is a C-set.
5. Every Zr-set is a Cr-set.
6. Every Zr-set is a Cr*-set.
7. Every Zr*-set is a C-set.
8. Every Zr*-set is a Cr-set.
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9. Every Zr*-set is a Cr*-set.
10. Every Zr*-set is a Z-set.
11. Every Zr*-set is a Zr-set.
12. Every A-set is a Z-set.
13. Every A-set is a Zr-set.
14. Every Z-set is a Zr-set.
15. Every t-set is a Zr-set.
Remark 6.3. The converses of the above are not true may be seen by the following examples.
Example 6.4. Let X={a,b,c}with τ={ϕ,{a},X}. Let A={a,b}. Then A is a C-set, Cr-set and Cr*-set but A is not a
Z-set.
Example 6.5. Let X={a,b,c}with τ={ϕ,{b},{a,b},X}. Let A={b}. Then A is a C-set, Cr-set and Cr*-set but A is
not a Zr-set.
Example 6.6. Let X={a,b,c}with τ={ϕ,{b,c},X}. Let A={a}. Then A is a C-set, Cr-set and Cr*-set but A is not a
Zr*-set.
Example 6.7. Let X={a,b,c}with τ={ϕ,{a},{c},{a,c},X}. Let A={b}. Then A is a Z-set, Zr-set but A is not a Zr*-
set.
Example 6.8. Let X={a,b,c}with τ={ϕ,{a},{b},{a,b},{b,c},X}. Let A={c}. Then A is a Z-set, Zr-set but A is not
a A-set.
Example 6.9. In example 6.6, Let A={a,b}. Then A is Zr-set but A is not a Z-set.
Example 6.10. In example 6.7, Let A={a}. Then A is Zr-set but A is not a t-set.
Remark 6.11. The concepts of Z-set and α*
-set are independent of each other as seen from the following
example.
Example 6.12. Let X={a,b,c}with τ={ϕ,{a,c},X}. Let A={a,c} is a Z-set but not a α*
-set and let A={a,b} is a α*
-
set but not a Z-set.
Remark 6.13. The concepts of Zr*-set and t-set are independent of each other as seen from the following
example.
Example 6.14. In example 6.5. Let A={b} is a Zr*-set but not a t-set and let A={a} is a t-set but not a Zr*-set.
Remark 6.15. The concepts of Zr*-set and α*
-set are independent of each other as seen from the following
example.
Example 6.16. In Example 6.12, Let A={a.c} is a Zr*-set but not a α*
-set and let A={b} is a α*
-set but not a Zr*-
set.
Proposition 6.17. If S is a gr*
-open set, then
i) S is a Z-set.
ii) S is a Zr-set.
iii) S is a Zr*-set.
Remark 6.18. The converse of the above proposition are not true may be seen by the following examples.
Example 6.19. In example 6.7. Let A={a,b} is a Z-set, Zr-set and Zr*-set but not a gr*
-open set.
The above discussions are summarized in the following diagram
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Proposition 6.20. Let A and B are Z-sets in X. Then A∩B is a Z-set in X.
Proof. Since A, B are Z-sets. Let A=L1∩M1, B=L2∩M2 where L1,L2 are gr*
-open sets and M1 , M2 are t-sets.
Since intersection of two gr*
-open sets is gr*-open sets and intersection of t-sets is t-set it follows that A∩B is a
Z-set in X.
Remark 6.21. a) The union of two Zr-sets need not be a Zr-set.
b) Complement of a Zr-sets need not be a Zr-set.
Example 6.22. In example 6.4.
a) A={a} and B={b} are Zr-sets but A∪B={a,b} is not a Zr-set.
b)X−{b}={a,c} is not a Zr-set.
Proposition 6.23. Let A and B be Zr-sets in X. Then A∩B is also a Zr-sets.
Remark 6.24. The union of two Zr*-sets is also a Zr*-sets and the complement of a Zr*-set need not be a Zr*-sets
follows from the following example.
Example 6.25. In example 6.6. Let A={b} and B={c} are Zr*-sets and A∪B={b,c} is also a Zr*-sets. Let
X−{b}={a,b} is not a Zr*-set.
VII. DECOMPOSITION OF GR*-CONTINUITY
Definition 7.1. A function f: X→Y is said to be
i) Z-continuous if f-1
(V) is a Z-set for every open set V in Y.
ii) Zr-continuous if f-1
(V) is a Zr-set for every open set V in Y.
iii) Zr*-continuous if f-1
(V) is a Zr*-set for every open set V in Y.
Proposition 7.2.
i) Every Zr*-continuous function is Z-continuous.
ii) Every Zr*-continuous function is Zr-continuous.
iii) Every Z-continuous function is Zr-continuous.
Proof. Follows from 6.2 and 7.1
Remark 7.3. Converses of the above proposition are not true may be seen by the following example.
Example 7.4.
The above discussions are summarized in the following implications..
10. On Decomposition of gr*
- closed set in Topological Spaces
www.ijmsi.org 54 | P a g e
Defintion 7.5. A map f: X→Y is said to be
i) Z-open if f(U) is a Z-set in Y for each open set V in X.
ii) Zr-open if f(U) is a Zr-set in Y for each open set V in X.
iii) Zr*-open if f(U) is a Zr*-set in Y for each open set V in X.
Definition 7.6. A map f: X→Y is said to be
i) contra Z-continuous if f-1
(V) is a Z-set for every closed set V in Y.
ii) contra Zr-continuous if f-1
(V) is a Zr-set for every closed set V in Y.
i) contra Zr*-continuous if f-1
(V) is a Zr*-set for every closed set V in Y.
Theorem 7.7. A subset A of X is
i) gr*
-open if only if it is both g*
-open and a Z-set in X.
ii) gr*
-open if only if it is both g*
-open and a Zr -set in X.
iii) gr*
-open if only if it is both rg-open and a Zr*-set in X.
Proof. Necessity: obvious
Sufficiency: Assume that A is both g*
-open and a Z-set in X. By assumption, A is a Z-set in X implies
A=L∩M, where L is gr*
-open and M is a t-set. Let F be a g-closed such that F⊂A, since A is gr*
-open, F⊂A
implies F⊂ rint(A)⊂ int(A). Then A is gr*
-open and F⊂A⊂L implies F⊂int(A). Hence
F⊂int(L)∩int(M)=int(L∩M)=int(A). Hence A is gr*
-open.
ii)Necessity: obvious
sufficiency: Assume that A is both g*
-open and a Zr-set in X. By assumption, A is a Zr-set in X implies
A=L∩M, where L is gr*
-open and M is a α*
-set. Let F be a g-closed such that F⊂A, since A is gr*
-open, F⊂A
implies F⊂ rint(A)⊂ int(A). Then A is gr*
-open and F⊂A⊂L implies F⊂int(A). Hence
F⊂int(L)∩int(M)=int(L∩M)=int(A). Hence A is gr*
-open.
iii)Necessity: obvious
sufficiency: Assume that A is both g*
-open and a Zr*-set in X. By assumption, A is a Zr*-set in X implies
A=L∩M, where L is gr*
-open and M is a A-set. Let F be a r-closed such that F⊂A, since A is gr*
-open, F⊂A
implies F⊂ rint(A)⊂ int(A). Then A is gr*
-open and F⊂A⊂L implies F⊂int(A). Hence
F⊂int(L)∩int(M)=int(L∩M)=int(A). Hence A is gr*
-open.
Theorem 7.8. A mapping f: X→Y is
i) gr*
-continuous if and only if it is both g*
-continuous and Z-continuous.
ii) gr*
-continuous if and only if it is both g*
-continuous and Zr-continuous.
iii) gr*
-continuous if and only if it is both rg-continuous and Zr*-continuous.
Proof. Follows from theorem 7.7.
Theorem 6.35. A map f: X→Y is
i) gr*
-open if and only if it is both g*
-open and Z-open.
ii) gr*
-open if and only if it is both g*
-open and Zr-open.
iii) gr*
-open if and only if it is both rg-open and Zr*-open.
Proof. Follows from theorem 7.7.
Theorem 6.36. A mapping f: X→Y is
i) contra gr*
-open if and only if f is both contra g*
-continuous and contra Z-continuous.
ii) contra gr*
-open if and only if f is both contra g*
-continuous and contra Zr-continuous.
iii) contra gr*
-open if and only if f is both contra rg-continuous and contra Zr*-continuous.
Proof. Follows from theorem 7.7.
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