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.
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.
In this paper, we introduce the concepts of πgθ-closed map, πgθ-open map, πgθ-
homeomorphisms and πgθc-homeomorphisms and study their properties. Also, we discuss its relationship
with other types of functions.
Mathematics Subject Classification: 54E55
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The 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.
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.
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological SpacesIOSR Journals
In this paper we introduce and study the concept of a new class of closed sets called (𝜏𝑖, 𝜏𝑗)− regular generalized b- closed sets (briefly(𝜏𝑖, 𝜏𝑗)− rgb-closed) in bitopological spaces.Further we define and study new neighborhood namely (𝜏𝑖, 𝜏𝑗)− rgb- neighbourhood (briefly(𝜏𝑖, 𝜏𝑗)− rgb-nhd) and discuss some of their properties in bitopological spaces. Also, we give some characterizations and applications of it.
This document 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.
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.
In this paper, we introduce the concepts of πgθ-closed map, πgθ-open map, πgθ-
homeomorphisms and πgθc-homeomorphisms and study their properties. Also, we discuss its relationship
with other types of functions.
Mathematics Subject Classification: 54E55
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The 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.
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.
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological SpacesIOSR Journals
In this paper we introduce and study the concept of a new class of closed sets called (𝜏𝑖, 𝜏𝑗)− regular generalized b- closed sets (briefly(𝜏𝑖, 𝜏𝑗)− rgb-closed) in bitopological spaces.Further we define and study new neighborhood namely (𝜏𝑖, 𝜏𝑗)− rgb- neighbourhood (briefly(𝜏𝑖, 𝜏𝑗)− rgb-nhd) and discuss some of their properties in bitopological spaces. Also, we give some characterizations and applications of it.
This document 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.
Strong Semiclosed Sets in Topological Spacesijtsrd
The document introduces and defines strong semiclosed sets in topological spaces. Key points:
- A subset S is defined as strong semiclosed if S = A∩F, where A is regular open and F is closed.
- It is shown that regular open and closed sets are strong semiclosed, but the converses do not hold. The finite union of strong semiclosed sets need not be strong semiclosed.
- Several properties of strong semiclosed sets are proved, including that strong semiclosed sets are semiclosed, β-closed, locally closed, D(c,p) sets, and simply open.
In general topology many strong and weak forms of open and closed sets have been defined
and studied. Govindappa Navalagi introduced the concept of semi α-open sets which is a weaker form of
α-open sets. Semi*α-open set is defined analogously by replacing the closure operator by the
generalized closure operator due to Dunham in the definition of semi α-open sets. In this paper we
introduce a new class of sets, namely semi*α-closed sets, as the complement of semi*α-open sets. We
find characterizations of semi*α-closed sets. We also define the semi*α-closure of a subset. Further we
investigate fundamental properties of the semi*α-closure. We define the semi*α-derived set of a subset
and study its properties.
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
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.
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.
μ-πrα Closed Sets in Bigeneralized Topological SpacesIJERA Editor
The aim of the paper is to introduce the concept of μ(m,n)-πrα closed sets in bigeneralized topological spaces and study some of their properties. We also introduce the notion of μ(m,n)-πrα continuous function and μ(m,n)-πrα T1/2 spaces on bigeneralized topological spaces and investigate some of their properties. Mathematics subject classification: 54A05, 54A10
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.
Compatible Mapping and Common Fixed Point TheoremIOSR Journals
This document presents a common fixed point theorem for compatible mappings. It begins with definitions of commuting, weakly commuting, and compatible mappings. It then states the main theorem - that if mappings P, Q, S, T satisfy certain conditions, including being compatible and a contraction-type inequality, then they have a unique common fixed point. The proof of the theorem is presented, showing that the mappings converge to a single point z, which is proven to be the unique common fixed point. A corollary is also presented as an extension of the main result.
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.
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.
This document outlines a seminar on motivic Hall algebras. The seminar will cover topics including Grothendieck rings of varieties, motivic invariants like Euler characteristics and Hodge polynomials, algebraic spaces, and stacks. The main focus will be T. Bridgeland's paper on motivic Hall algebras, which uses motives and stacks in its construction of motivic Hall algebras.
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
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
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
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 Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document introduces and studies the concepts of πgr-homeomorphisms and πgrc-homeomorphisms between topological spaces. It begins by providing definitions of related concepts such as πgr-closed maps, πgr-continuous maps, and πgr-irresolute maps. It then defines πgr-homeomorphisms as bijections that are both πgr-continuous and πgr-open, and πgrc-homeomorphisms as bijections whose inverse images are πgr-closed sets. Several properties and characterizations of these maps are established. It is shown that πgr-homeomorphisms and πgrc-homeomorphisms
Strong Semiclosed Sets in Topological Spacesijtsrd
The document introduces and defines strong semiclosed sets in topological spaces. Key points:
- A subset S is defined as strong semiclosed if S = A∩F, where A is regular open and F is closed.
- It is shown that regular open and closed sets are strong semiclosed, but the converses do not hold. The finite union of strong semiclosed sets need not be strong semiclosed.
- Several properties of strong semiclosed sets are proved, including that strong semiclosed sets are semiclosed, β-closed, locally closed, D(c,p) sets, and simply open.
In general topology many strong and weak forms of open and closed sets have been defined
and studied. Govindappa Navalagi introduced the concept of semi α-open sets which is a weaker form of
α-open sets. Semi*α-open set is defined analogously by replacing the closure operator by the
generalized closure operator due to Dunham in the definition of semi α-open sets. In this paper we
introduce a new class of sets, namely semi*α-closed sets, as the complement of semi*α-open sets. We
find characterizations of semi*α-closed sets. We also define the semi*α-closure of a subset. Further we
investigate fundamental properties of the semi*α-closure. We define the semi*α-derived set of a subset
and study its properties.
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
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.
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.
μ-πrα Closed Sets in Bigeneralized Topological SpacesIJERA Editor
The aim of the paper is to introduce the concept of μ(m,n)-πrα closed sets in bigeneralized topological spaces and study some of their properties. We also introduce the notion of μ(m,n)-πrα continuous function and μ(m,n)-πrα T1/2 spaces on bigeneralized topological spaces and investigate some of their properties. Mathematics subject classification: 54A05, 54A10
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.
Compatible Mapping and Common Fixed Point TheoremIOSR Journals
This document presents a common fixed point theorem for compatible mappings. It begins with definitions of commuting, weakly commuting, and compatible mappings. It then states the main theorem - that if mappings P, Q, S, T satisfy certain conditions, including being compatible and a contraction-type inequality, then they have a unique common fixed point. The proof of the theorem is presented, showing that the mappings converge to a single point z, which is proven to be the unique common fixed point. A corollary is also presented as an extension of the main result.
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.
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.
This document outlines a seminar on motivic Hall algebras. The seminar will cover topics including Grothendieck rings of varieties, motivic invariants like Euler characteristics and Hodge polynomials, algebraic spaces, and stacks. The main focus will be T. Bridgeland's paper on motivic Hall algebras, which uses motives and stacks in its construction of motivic Hall algebras.
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
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
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
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 Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document introduces and studies the concepts of πgr-homeomorphisms and πgrc-homeomorphisms between topological spaces. It begins by providing definitions of related concepts such as πgr-closed maps, πgr-continuous maps, and πgr-irresolute maps. It then defines πgr-homeomorphisms as bijections that are both πgr-continuous and πgr-open, and πgrc-homeomorphisms as bijections whose inverse images are πgr-closed sets. Several properties and characterizations of these maps are established. It is shown that πgr-homeomorphisms and πgrc-homeomorphisms
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
Contra qpi continuous functions in ideal bitopological spacesAlexander Decker
This academic article discusses contra qpI-continuous functions in ideal bitopological spaces. It begins with preliminaries on bitopological spaces, ideal topological spaces, and related concepts. It then defines contra qpI-continuous functions and establishes some of their properties, such as their relationship to contra qI-continuous functions and qpI-continuous functions. It proves several theorems about contra qpI-continuous functions and their images, including that the image of a qpI-connected space under a contra qpI-continuous function is connected. The article concludes by discussing contra qpI-irresolute mappings.
This document introduces and studies the concept of ˆ-closed sets in topological spaces. Some key points:
1. ˆ-closed sets are defined as sets whose δ-closure is contained in any semi-open set containing the set.
2. It is shown that ˆ-closed sets lie between δ-closed sets and various other classes like δg-closed and ω-closed sets.
3. Several characterizations of ˆ-closed sets are provided in terms of properties of the difference between the δ-closure of the set and the set itself.
4. The concept of the ˆ-kernel of a set is introduced, defined as the intersection of all ˆ-
This 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.
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
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.
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
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.
γ Regular-open sets and γ-extremally disconnected spacesAlexander Decker
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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.
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International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The aim of this paper is to introduce pgrw-closed maps and pgrw*-closed maps and to obtain some of their properties. In section 3 pgrw-closed map is defined and compared with other closed maps. In section 4 composition of pgrw-maps is studied. In section 5 pgrw*-closed maps are defined.
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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.
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number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
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including those with tonal differences. The proposed method offers a practical and efficient solution for
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A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively.
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other with an adaptive one. In the latter, we have learned the three types of detectors CA, SO, and GO-CFAR. This research
aims to apply intelligent techniques to improve detection performance in a nonhomogeneous environment using standard
CFAR detectors. The objective is to maintain the false alarm probability and enhance target detection by combining
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OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints.
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A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal argument in its original form to establish the cardinality of K between that of (N,R) respectively.
Moving Target Detection Using CA, SO and GO-CFAR detectors in Nonhomogeneous ...mathsjournal
Modernization of radar technology and improved signal processing techniques are necessary to improve detection systems in complex situations. A fundamental problem in radar systems is to automatically detect targets while maintaining a
desired constant false alarm probability. This work studies two detection approaches, the first with a fixed threshold and the
other with an adaptive one. In the latter, we have learned the three types of detectors CA, SO, and GO-CFAR. This research
aims to apply intelligent techniques to improve detection performance in a nonhomogeneous environment using standard
CFAR detectors. The objective is to maintain the false alarm probability and enhance target detection by combining
intelligent techniques. With these objectives in mind, implementing standard CFAR detectors is applied to nonhomogeneous
environment data. The primary focus is understanding the reason for the false detection when applying standard CFAR
detectors in a nonhomogeneous environment and how to avoid it using intelligent approaches
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints
Modified Alpha-Rooting Color Image Enhancement Method on the Two Side 2-D Qua...mathsjournal
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functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
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1. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 1, March 2015
27
ON β-NORMAL SPACES
1
o. Ravi, 2
i. Rajasekaran, 3
s. Murugesan And 4
a. Pandi
1;2
Department of Mathematics,P. M. Thevar College, Usilampatti, Madurai District,
Tamil Nadu, India.
3
Department of Mathematics, Sri S. Ramasamy Naidu Memorial College, Sattur-626 203,
Tamil Nadu, India.
4
Department of Mathematics, The Madura College, Madurai, Tamil Nadu, India.
ABSTRACT
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, p-
normal 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.
1. INTRODUCTION
First step in normality was taken by Viglino [32] who de_ned semi normal spaces. Then Singal
and Arya [28] introduced the class of almost normal spaces and proved that a space is 02010
Mathematics Subject Classi_cation: Primary : 54D10, Secondary : 54D15, 54A05, 54C08.
Key words and phrases. p-normal space, s-normal space, β-normal space, gβ-closed function, β-
gβclosed function, separation axioms. normal if and only if it is both a semi-normal space and an
almost normal space. Normality is an important topological property and hence it is of
signi_cance both from intrinsic interest as well as from applications view point to obtain
factorizations of normality in terms of weaker topological properties. In recent years, many
authors have studied several forms of normality [10, 12, 14, 24]. On the other hand, the notions of
p-normal spaces and s-normal spaces were introduced by Paul and Bhattacharyya [27]; and
Maheshwari and Prasad [17], respectively.
Levine [16] initiated the investigation of g-closed sets in topological spaces, since then many
modi_cations of g-closed sets were de_ned and investigated by a large number of topologists [5,
7, 10, 25]. In 1996, Maki et al [19] introduced the concepts of gp-closed sets and Arya and Nour
[4] introduced the concepts of gs-closed sets. The purpose of this paper is to study the class of
2. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 1, March 2015
28
normal spaces, namely β-normal spaces, which is a generalization of the classes of p-normal
spaces and s-normal spaces. The relations among β-normal spaces, p-normal spaces and s-normal
spaces and also properties of β-normal spaces are investigated. Moreover, we study the forms of
generalized _-closed functions. We obtain properties of these forms of generalized β-closed
functions and preservation theorems.
Spaces always mean topological spaces on which no separation axioms are assumed unless
explicitly stated and (or simply denotes a function f of a
space into a space . Let A be a subset of a space X. The closure and the interior of A
are denoted by cl(A) and int(A) respectively.
De_nition 2.1. A subset A of a space X is called
(1) regular open [29] if A = int(cl(A));
(2) β-open [22] if A int(cl(int(A)));
(3) semi-open [15] if A cl(int(A));
(4) β-open [1] if A cl(int(cl(A)));
(5) preopen [21] or nearly open [11] if A int(cl(A)).
It is shown in [22] that the class of _-open sets is a topology and it is stronger than given topology
on X.
The complement of an α-open (resp. semi-open, preopen, β-open, regular open) set is called α-
closed [20] (resp. semi-closed [9], preclosed [21], β-closed [1], regular closed [29]).
The intersection of all α-closed (resp. semi-closed, preclosed, β-closed) sets containing A is
called the α-closure (resp. semi-closure, preclosure, α-closure) of A and is denoted by αcl(A)
(resp. s-cl(A), p-cl(A), β-cl(A)).
Dually, the α-interior (resp. semi-interior, preinterior, β-interior) of A, denoted by β-int(A) (resp.
sint(A), pint(A), β-int(A)), is defined to be the union of all α-open (resp. semi-open, preopen, β-
open) sets contained in A.
The family of all β-open (resp. β-closed, α-open, regular open, regular closed, semi-open,
preopen) sets of a space X is denoted by βO(X) (resp. βC(X), βO(X), RO(X), RC(X), SO(X),
PO(X)). The family of all β-open sets of X containing a point x is denoted by βO(X, x).
Lemma 2.2. [2] Let A be a subset of a space X and x 2 X. The following properties hold for β-
cl(A):
(1) x € β-cl(A) if and only if A∩U 6= _ for every U € βO(X) containing x;
(2) A is β-closed if and only if A = β-cl(A);
(3) β-cl(A) β-cl(B) if A B;
3. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 1, March 2015
29
(4) β-cl(_-cl(A)) = β-cl(A);
(5) β-cl(A) is β-closed.
De_nition 2.3. A space X is said to be prenormal [26] or p-normal [27] (resp. s-normal [17]) if for
any pair of disjoint closed sets A and B, there exist disjoint preopen (resp. semi-open) sets U and
V such that A U and B V.
De_nition 2.4. A subset A of a space is said to be g-closed [16] (resp. gs-closed [4], gp-
closed [19]) if cl(A) U (resp. s-cl(A) U, p-cl(A) U) whenever A_U and U € .
The complement of g-closed (resp. gs-closed, gp-closed) set is said to be g-open (resp. gs-open,
gp-open).
Definition 2.5. A subset A of a space is said to be sg-closed [5] (resp. pg-closed [6]) if s-
cl(A) U (resp. p-cl(A) U) whenever A U and U € SO(X) (resp. U € PO(X)).
The complement of sg-closed (resp. pg-closed) set is said to be sg-open (resp. pg- pen).
3. β-NORMAL SPACES
Definition 3.1. [18] A space X is said to be β-normal if for any pair of disjoint closed sets A and
B, there exist disjoint β-open sets U and V such that A U and B V.
Remark 3.2. The following diagram holds for a topological space .
None of these implications is reversible as shown by the following Examples.
s-normal.
4. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 1, March 2015
30
For the other implications the Examples can be seen in [11].
Theorem 3.4. For a space X the following are equivalent :
(1) X is β-normal,
(2) For every pair of open sets U and V whose union is X, there exist _-closed sets A and B
such that A U, B V and A U B= X,
(3) For every closed set H and every open set K containing H, there exists a β-open setU
such that H U β-cl(U) K.
4. THE RELATED FUNCTIONS WITH β-NORMAL SPACES
Definition 4.1. A function f : X → Y is called
(1) pre β-open if f(U) 2 βO(Y) for each U €βO(X) [18];
5. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 1, March 2015
31
(2) pre β-closed if f(U) 2 βC(Y) for each U €βC(X) [18];
(3) almost β-irresolute if for each x in X and each β-neighbourhood V of f(x), β-cl(f-1
(V)) is a β-
neighbourhood of x.
Theorem 4.2. A function f : X → Y is pre β-closed if and only if for each subset A in Y and for
each _-open set U in X containing f-1
(A), there exists a β-open set V of Y containing A such that
f-1
(V) U.
6. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 1, March 2015
32
Theorem 4.8. If f : X → Y is an β-closed continuous surjection and X is normal, then Y is β-
normal.
Proof. Let A and B be disjoint closed sets of Y . Then f-1
(A) and f-1
(B) are disjoint closed sets of
X by the continuity of f. As X is normal, there exist disjoint open sets U and V in X such that f-
1
(A) U and f-1
(B) V . By Proposition 6 in [23], there are disjoint _-open sets G and H in
Y such that A G and B H. Since every _-open set is -open, G and H are disjoint β-
open sets containing A and B, respectively. Therefore, Y is β-normal.
5. GENERALIZED Β-CLOSED FUNCTIONS
7. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 1, March 2015
33
Definition 5.1. [31] A subset A of a space is said to be gβ-closed if β-cl(A) U
whenever A U and U € .
De_nition 5.2. A subset A of a space is said to be _g-closed if β-cl(A) _ U whenever A
U and U € βO(X).
The complement of βg-closed set is said to be βg-open.
Remark 5.3. The following diagram holds for any subset of a topological space X.
None of these implications is reversible as shown by the following Examples and the related
papers.
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For the other implications the examples can be seen in [4, 5, 6, 9, 19, 21].
Definition 5.7. A function f : X → Y is said to be
(1) β-closed if f(A) is β-closed in Y for each closed set A of X [1],
(2) βg-closed if f(A) is βg-closed in Y for each closed set A of X,
(3) gβ-closed if f(A) is gβ-closed in Y for each closed set A of X.
Definition 5.8. A function f : X → Y is said to be
(1) quasi β-closed if f(A) is closed in Y for each A € βC(X),
(2) β-βg-closed if f(A) is βg-closed in Y for each A € βC(X),
(3) β-gβ-closed if f(A) is gβ-closed in Y for each A € βC(X) [31],
(4) almost gβ-closed if f(A) is gβ-closed in Y for each A € RC(X).
Remark 5.9. The following diagram holds for a function f : → :
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Definition 5.12. A function f : X → Y is said to be β-gβ-continuous [30] if f-1
(K) is gβ-closed in
X for every K € βC(Y)
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13. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 1, March 2015
39
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