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
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.
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.
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 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.
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.
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.
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.
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.
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 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.
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.
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.
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 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, 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.
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological SpacesIOSR Journals
In this paper we introduce and study the concept of a new class of closed sets called (𝜏𝑖, 𝜏𝑗)− regular generalized b- closed sets (briefly(𝜏𝑖, 𝜏𝑗)− rgb-closed) in bitopological spaces.Further we define and study new neighborhood namely (𝜏𝑖, 𝜏𝑗)− rgb- neighbourhood (briefly(𝜏𝑖, 𝜏𝑗)− rgb-nhd) and discuss some of their properties in bitopological spaces. Also, we give some characterizations and applications of it.
International Journal of 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.
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.
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.
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 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.
Continuous And Irresolute Functions Via Star Generalised Closed SetsIJMERJOURNAL
ABSTRACT: In this paper, we introduce a new class of continuous functions called semi*δ-continuous function and semi* δ-irresolute functions in topological spaces by utilizing semi* δ-open sets and to investigate their properties.
On Decomposition of gr* - closed set in Topological Spacesinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
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.
In this paper we introduce a new class of sets known as
δ
ˆ
S –closed sets in ideal topological
spaces and we studied some of its basic properties and characterizations. This new class of sets lies
between –I–closed [19] sets and g–closed sets, and its unique feature is it forms topology and it is
independent of open sets.
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-
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.
Abstract: In this paper, we define and study about a new type of generalized closed set called, g∗s-closed set.Its relationship with already defined generalized closed sets are also studied
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 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 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, 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.
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological SpacesIOSR Journals
In this paper we introduce and study the concept of a new class of closed sets called (𝜏𝑖, 𝜏𝑗)− regular generalized b- closed sets (briefly(𝜏𝑖, 𝜏𝑗)− rgb-closed) in bitopological spaces.Further we define and study new neighborhood namely (𝜏𝑖, 𝜏𝑗)− rgb- neighbourhood (briefly(𝜏𝑖, 𝜏𝑗)− rgb-nhd) and discuss some of their properties in bitopological spaces. Also, we give some characterizations and applications of it.
International Journal of 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.
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.
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.
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 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.
Continuous And Irresolute Functions Via Star Generalised Closed SetsIJMERJOURNAL
ABSTRACT: In this paper, we introduce a new class of continuous functions called semi*δ-continuous function and semi* δ-irresolute functions in topological spaces by utilizing semi* δ-open sets and to investigate their properties.
On Decomposition of gr* - closed set in Topological Spacesinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
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.
In this paper we introduce a new class of sets known as
δ
ˆ
S –closed sets in ideal topological
spaces and we studied some of its basic properties and characterizations. This new class of sets lies
between –I–closed [19] sets and g–closed sets, and its unique feature is it forms topology and it is
independent of open sets.
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-
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.
Abstract: In this paper, we define and study about a new type of generalized closed set called, g∗s-closed set.Its relationship with already defined generalized closed sets are also studied
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
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
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
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.
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.
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.
This document discusses generalized closed sets in topological spaces. It begins by introducing several types of generalized closed sets that have been defined in previous literature, such as g-closed sets, sg-closed sets, gs-closed sets, etc. It then defines a new type of generalized closed set called a g*s-closed set, which is a subset A such that the semi-closure of A is contained in every g-open set containing A. Examples are provided to illustrate g*s-closed sets. Properties of g*s-closed sets are discussed, such as every semi-closed set being g*s-closed, but the converse is not true. The relationship between g*s-closed sets and other
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
Fibrewise near compact and locally near compact spacesAlexander Decker
This document defines and studies new concepts of fibrewise topological spaces over a base set B, namely fibrewise near compact and fibrewise locally near compact spaces. These are generalizations of near compact and locally near compact topological spaces. Key definitions include:
1. Fibrewise near compact spaces, where the projection map is a "near proper" function and each fibre is near compact.
2. Fibrewise locally near compact spaces, where each point has a neighbourhood whose closure is near compact.
3. Relationships between these concepts and some fibrewise near separation axioms are also studied.
Fibrewise near compact and locally near compact spaces
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.
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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.
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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 .
(g f) (V) is πgr - closed in Z for the closed set V of X .
(g f) 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 .
(g f) (V) is πgr - closed in Z for the closed set V of X .
(g f) 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
(g f) 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.
(g f) (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-1are 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.
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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 .
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[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.
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Hence (gof) (U) = g([f(U)] is πgr- open in Z for every open set U of X.
(g f) 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 :
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We know that the composition of two πgrch(X,τ) is again a πgrch(X,τ).ie, For all f, g πgrch(X,τ), g f
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
= (f k1 f-1) (f k2 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))
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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.