In the category of fuzzy topological space the research and investigations show that the methods of algebraic topology are almost never used. The reason for this is that there is no given appropriate homotopy concept. Actually, there exist three different approach to define homotopy concept in fuzzy topological spaces. Therefore,homology theory in the category of fuzzy topological space is introduced.

Zadeh (1965), introduced the basic idea of a fuzzy set as an extension of classical set theory. In 1968, Chang was the first to introduce the concept of a fuzzy topology. Kubiak and Sostak (1985) introduced the fundamental concept of a fuzzy topological structure, as an extension of both crisp topology and fuzzy topology, in the sense that not only the objects are fuzzified, but also the axiomatics. Also, Sostak gave some rules and showed how such an extension can be realized. The purpose of this book is to introduce and study some new definitions of fuzzy sets, continuous mappings and separation axioms On Fuzzy Topological Spaces In Sostak's Sence.

We have carefully put this book for those interested in the different branches of fuzzy mathematics that are used in a variety of applications in other scientific fields such as engineering, computer science, biology, geography and others. In recent years, there is an increased interest in the concept of fuzzy sets that led to the concept of fuzzy logic which is used in several areas, such as control-theory, game-theory, economics, statistics, data management, artificial intelligence, automated reasoning etc. What distinguishes this book is that it contains many examples not available before, and characterized by their lengthy accounts. The origin of this difficulty is the nature of fuzzy sets provided in 1965 by Lotfi Zadeh which are more complicated than the ordinary set. Our system in this work depends on using the idea of the generalized closed sets studied by Levin, in 1970, which I identified it on the fuzzy sitting on different types. It is a powerful mean to describe the topological spaces that meet weak separation axioms applicable in computer science. Throughout this work, we elucidate new born results, theories and fuzzy continuous mappings in [0,1]-topological spaces.

In this book, the notion of noncompact covering properties, viz., metacompactness, subparacompactness, submetacompactness and para-Lindelofness in L-topological spaces are introduced using quasi coincidence relation and a-Q-covers. Moreover the characterizations of these properties in the weakly induced L-topological spaces are obtained. Further the invariance of these properties under various kinds of fuzzy maps is also studied. Some brief sketches regarding the related topics covering dimension and finitistic spaces in L-topological spaces is also provided.

The purpose of this book is to introduce some new definitions of separation axioms in supra fuzzy topological spaces using the ideas of Ali [8]. Some of their equivalent formulations along with various new characterizations and results concerning the existing ones are presented here. Our criterion for definitions has been preserving as much as possible the relation between the corresponding separation properties for supra fuzzy topological spaces. Moreover, it will be seen that the definitions of these axioms are ‘good extensions’ in the sense of Lowen [60]. We aim to develop theories of supra fuzzy T0, supra fuzzy T1, supra fuzzy T2(Hausdorff), supra fuzzy SFR(supra fuzzy regular), supra fuzzy SFN(supra fuzzy normal), supra fuzzy R0- and supra fuzzy R1- separation axioms analogous to its counter part in ordinary topology. We also prove that all the concepts are hereditary, productive and projective.

Loop space recognition is well understood in the context of traditional topology, however many principles for loop space characterization can be generalized to contexts besides topological spaces. Quillen introduced model categories as categories that have properties sufficient to define a homotopy theory consistent with the classical notion from topology. These categories can be understood by a class of morphisms called weak equivalences that yield isomorphism in the defined homotopy theory. When a model category possesses sufficiently rich combinatorial properties, the class of weak equivalences can be enlarged in a localization of the category. This work explores recognition principles, consistent with classical loop space recognition, in the context of model categories with localization. Analogues to the classical topological loop functors are provided in these categories with a corresponding recognition principle.

In 1965, L.A. Zadeh introduced the concept of Fuzzy Set as an attempt to mathematically handle those situations which are inherently vogue, imprecise or fuzzy in nature. The advent of Fuzzy Set Theory has led to the development of many areas of Mathematics and Sciences. It has become a concern and a new tool for mathematicians working in many different disciplines of Mathematics. After the introduction of "Fuzzy Topology" by Chang C.L. many concepts of "General Topology" have been fuzzified successfully by many researchers. The present book entitled "Fuzzy R0 and R1 Properties" is devoted to finding out some new R1-concepts for fuzzy topological spaces. Besides some existing concepts of fuzzy R0, R1, T0, T1, T2 and regular topological spaces are recalled, and studied in detail. Interrelations among various separation axioms of fuzzy topological spaces are discussed. This book will be helpful for the undergraduate and graduate students, particularity for those who are studying Fuzzy Mathematics.

Achievements And Awards In Abroad: YOUNG SCIENTIST (DST). MALAYASIA, 2010-2011. 1. International Conference On Mathematics And Statistics Held At Kuala-Lampur, Malayasia In 3 - 5, November 2010 - 2011. Selected As Young Scientist (DST), SENIOR SCIENTIST (DST). LONDON, U.K. 2011-2012. 2. World Science Congress (LONDON) Is An International Conference Of Applied And Engineering Mathematics, Selected As A Senior Scientist Held At London, U.K. In July 6 - 9, 2011 – 2012. (DST). KEY FEATURES: Each topic is treated in systematic, logical and lucid manner. The Scope and depth of coverage of each topic are strictly according to the world wide fuzzy topological space syllabus. Concepts are introduced in a sequential way with detailed explanations and illustrations. This book is entirely useful in fuzzy logic Programme and Fuzzy Topological Spaces.

The problems associated in the present research work which are taken care of are related to the study of fixed point theorems in semi Hausdorff space, uniform space and Banach space along with studying the applications of fixed point theorem in economics and analyzing the theory of fractals and application of fixed point theory. In this work, the introductory part of fractal theory, the important applications of fractal in image compression and a model of fixed point theorem in dynamical system is taken into consideration keeping in view its application in various areas like planetary motion, stock market prices, meteorological data etc. Also theorem of maximal elements in Banach space and existence of maximal elements and equilibrium for pair of maps in

Each topic is treated in systematic logical and lucid manner. The Scope and depth of coverage of each topic are strictly according to the world wide fuzzy topological space syllabus.Concepts are introduced in a sequential way with detailed explanations. Dr. Sadanand N. Patil is a working as Assistant Professor and H.O.D in the D/o. Mathematics, KLS’s, VDRIT, HALIYAL. Karnataka, INDIA. Affiliated to VTU, BELGAUM. Since last 5 yrs he is teaching to UG & PG students of engineering stream. He has obtained M.Sc. (Mathematics) in the year 2002, PGDCA, in the year 2003 and He has obtained Ph.D. in Mathematics in the year 2009. Under the K U, Dharwad. The title of the thesis is SOME RECENT ADVANCE TOPICS IN TOPOLOGY, (FUZZY TOPOLOGICAL SPACES). Presently he is guiding to 6 candidates for Ph.D. in the field of FUZZY LOGIC Using FUZZY TOPOLOGICAL SPACES. As young scientist (DST) presented 2 papers in International Conference On Mathematics And Statistics Held At Kuala-Lampur, Malayasia (3 -5, November 2010). As senior scientist (DST) presented 2 papers in World Science Congress (LONDON) Is An IAENG , London, UK (July 6 - 9, 2011.) He Presented 5 Papers International Conference in Abroad, Presented 42 Papers In Various National and International Conferences and also Published 40 Papers In Various International Reputed Journals.

When planning experiments it is very important to know the amount of information that we obtain from their realization. It is known that a measure of information is entropy. A usual mathematical model of experiment in the classical information theory is a measurable partition. Partitions are standardly defined in the context of classical sets. It has appeared however, that for a solution of real problems are partitions defined by means of the concept of fuzzy sets more appropriate. The book contains a review of results of author related to the study of entropy in the fuzzy case. These results represent fuzzy generalizations of some concepts from the classical probability theory. The entropy of fuzzy partition can be considered as a measure of information of experiment, the results of which are fuzzy events. The concept of entropy of a fuzzy partition is exploited to define the entropy of fuzzy dynamical systems. Subsequently an ergodic theory for these systems is proposed. Presented results include the corresponding results from the classical ergodic theory as particular cases. The publication might be useful for specialists who deal in their research work with similar issues.

This book describes a new concept of fine topological space. The collection of fine open sets contains all semi-open, pre-open, ?-open,?-open etc. sets which are used in defining the lighter concepts of continuity. With this wider class of fine sets the authors have defined the continuity which includes several other continuities already defined. Csaszar A. has introduced the concept of generalized topological space in 2002. A fine topological space is a special case of generalized topological space, but it may be noted that the concept of fine space is based on a topological space. The notions of several continuous functions in topological spaces and fuzzy topological spaces are widely developed which are used extensively in many practical and engineering problems. The concept of continuity and homeomorphism are used in many fields such as in Chemistry, Computer Science, Physics, Zoology, Quantum physics, Quantum mechanics etc. Now, there is a wide scope of study this concept in view of fuzzy topological space.

This book deals with a concise study of convergence in intuitionistic fuzzy n-normed linear spaces. This book mainly contains the author's own research work in the area of lacunary ideal convergence. Fuzzy normed spaces have been an increasingly popular area of mathematical research in recent times, both in terms of theory and applications. But the availability of books in the area of fuzzy normed spaces is very rare. This book provides a good discussion on the development of both fuzzy and intuitionistic fuzzy set theory. The transition from fuzzy normed linear spaces to intuitionistic fuzzy n-normed linear spaces has been presented systematically. Anybody interested in the theory or application of fuzzy or intuitionistic fuzzy normed spaces will find this book more than useful. The book is written in such a way that mathematical prerequisites are minimum. Since the main subject of study in this book is a generalisaton of the concept of usual convergence, so all the related results in convergence have been incorporated in the book. This book may be used as a ready reference for an up to date account of results in the theory of fuzzy/intuitionistic fuzzy normed linear spaces.

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, continuity, proximity, uniformity, and syntopogenous structures. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology. The notions of fuzzy sets and intuitionistic fuzzy sets used to generalize the topological spaces. Recently, Garcia and Rodabaugh could end some doubts around the term “intuitionistic” by giving it the new name “double”. Since that time, the new term is used to study the topological notions.

This short monograph offers a new vista of geometric topology from the standpoint of operator algebras. The common topic are functors from topological and geometric spaces to a category of operator algebras; such functors help to solve open problems in topology which are out of reach otherwise. Chapter 1 deals with new invariants of three-dimensional manifolds obtained from operator algebras. Chapter 2 links homology invariants of three-dimensional manifolds with the K-theory of Cuntz-Krieger algebras. Chapter 3 settles the Harvey conjecture for the mapping class group using ideas and methods of the noncommutative geometry.

In this book, we study the lattice valued intuitionistic fuzzy topological spaces, lattice valued intuitionistic fuzzy syntopogenous structures, lattice valued intuitionistic proximity and lattice valued intuitionistic fuzzy uniformity in Entourag approach. Also, we construct stratified intuitionistic L-topological structures from a given intuitionistic L-topological structures. Finally, we introduce a new intuitionistic LC-uniformity in term of covering approach

The purpose of this work is to investigate the topological nature of Fuzzy measure and relationship on the continuities between possibility density function and Possibility measure on compact topological spaces. The research deals with the concept of compactness of fuzzy topological spaces, continuities and regularity between Possibility measure and possibility density function of compact topological spaces and related ideas.

In my book, I give the analogy of the main results in one of Kevin Costello’s paper for open-closed topological conformal field theory. In other words, I show that there is a Batalin-Vilkovisky algebraic structure on the open-closed moduli space (moduli space of Riemann surface with boundary and marked points) , which is defined by Harrelson, Voronov and Zuniga, and the most important, there is a solution up to homotopy to the quantum master equation of that BV algebra if the initial condition is given, under the assumption that a new geometric chain theory gives rise to ordinary homology. This solution is expected to encode the fundamental chain of compactified open-closed moduli space, which is studied thoroughly by C.-C.Liu, as exactly in the closed case. We hope this result can give new insights into the mysterious two dimensional open-closed field theory.

It has been shown that the algebraic and topological study of automata theory played a key role in the development of fundamentals of computer science. This book demonstrates some algebraic and topological aspects of rough automata and fuzzy automata. Specifically, this text present the foundations of algebraic theory of rough automata which will attract the researchers and will help in finding some successful applications of rough automata theory. Ideas for future research and explorations are also provided. Students and researchers in computer science and mathematics will benefit from this work.

The concept of fuzzy set was introduced by Zadeh in his classical paper (Zadeh, 1965). Balasubramaniam and Sundaram defined generalized fuzzy closed sets in a fuzzy topological space and introduced certain types of fuzzy continuous functions between fuzzy topological spaces.(X, T) and (Y, S) denotes a fuzzy topological spaces.Also in this book I introduced different types of open sets and closed sets in fuzzy topological spaces.I think it will be very useful to co higher research in this field.

## homology theory in the category of fuzzy topological spaces в наличии / купить интернет-магазине

## Homology Theory in the Category of Fuzzy Topological Spaces

In the category of fuzzy topological space the research and investigations show that the methods of algebraic topology are almost never used. The reason for this is that there is no given appropriate homotopy concept. Actually, there exist three different approach to define homotopy concept in fuzzy topological spaces. Therefore,homology theory in the category of fuzzy topological space is introduced.

## A Study On Fuzzy Topological Spaces In Sostak's Sence

Zadeh (1965), introduced the basic idea of a fuzzy set as an extension of classical set theory. In 1968, Chang was the first to introduce the concept of a fuzzy topology. Kubiak and Sostak (1985) introduced the fundamental concept of a fuzzy topological structure, as an extension of both crisp topology and fuzzy topology, in the sense that not only the objects are fuzzified, but also the axiomatics. Also, Sostak gave some rules and showed how such an extension can be realized. The purpose of this book is to introduce and study some new definitions of fuzzy sets, continuous mappings and separation axioms On Fuzzy Topological Spaces In Sostak's Sence.

## Generalized Closed Fuzzy Sets in [0,1]-topological Spaces

We have carefully put this book for those interested in the different branches of fuzzy mathematics that are used in a variety of applications in other scientific fields such as engineering, computer science, biology, geography and others. In recent years, there is an increased interest in the concept of fuzzy sets that led to the concept of fuzzy logic which is used in several areas, such as control-theory, game-theory, economics, statistics, data management, artificial intelligence, automated reasoning etc. What distinguishes this book is that it contains many examples not available before, and characterized by their lengthy accounts. The origin of this difficulty is the nature of fuzzy sets provided in 1965 by Lotfi Zadeh which are more complicated than the ordinary set. Our system in this work depends on using the idea of the generalized closed sets studied by Levin, in 1970, which I identified it on the fuzzy sitting on different types. It is a powerful mean to describe the topological spaces that meet weak separation axioms applicable in computer science. Throughout this work, we elucidate new born results, theories and fuzzy continuous mappings in [0,1]-topological spaces.

## Topics in Fuzzy Topology

In this book, the notion of noncompact covering properties, viz., metacompactness, subparacompactness, submetacompactness and para-Lindelofness in L-topological spaces are introduced using quasi coincidence relation and a-Q-covers. Moreover the characterizations of these properties in the weakly induced L-topological spaces are obtained. Further the invariance of these properties under various kinds of fuzzy maps is also studied. Some brief sketches regarding the related topics covering dimension and finitistic spaces in L-topological spaces is also provided.

## Supra Fuzzy Topological Spaces

The purpose of this book is to introduce some new definitions of separation axioms in supra fuzzy topological spaces using the ideas of Ali [8]. Some of their equivalent formulations along with various new characterizations and results concerning the existing ones are presented here. Our criterion for definitions has been preserving as much as possible the relation between the corresponding separation properties for supra fuzzy topological spaces. Moreover, it will be seen that the definitions of these axioms are ‘good extensions’ in the sense of Lowen [60]. We aim to develop theories of supra fuzzy T0, supra fuzzy T1, supra fuzzy T2(Hausdorff), supra fuzzy SFR(supra fuzzy regular), supra fuzzy SFN(supra fuzzy normal), supra fuzzy R0- and supra fuzzy R1- separation axioms analogous to its counter part in ordinary topology. We also prove that all the concepts are hereditary, productive and projective.

## Loop Spaces in Motivic Homotopy Theory

Loop space recognition is well understood in the context of traditional topology, however many principles for loop space characterization can be generalized to contexts besides topological spaces. Quillen introduced model categories as categories that have properties sufficient to define a homotopy theory consistent with the classical notion from topology. These categories can be understood by a class of morphisms called weak equivalences that yield isomorphism in the defined homotopy theory. When a model category possesses sufficiently rich combinatorial properties, the class of weak equivalences can be enlarged in a localization of the category. This work explores recognition principles, consistent with classical loop space recognition, in the context of model categories with localization. Analogues to the classical topological loop functors are provided in these categories with a corresponding recognition principle.

## Fuzzy R0 and R1 Properties

In 1965, L.A. Zadeh introduced the concept of Fuzzy Set as an attempt to mathematically handle those situations which are inherently vogue, imprecise or fuzzy in nature. The advent of Fuzzy Set Theory has led to the development of many areas of Mathematics and Sciences. It has become a concern and a new tool for mathematicians working in many different disciplines of Mathematics. After the introduction of "Fuzzy Topology" by Chang C.L. many concepts of "General Topology" have been fuzzified successfully by many researchers. The present book entitled "Fuzzy R0 and R1 Properties" is devoted to finding out some new R1-concepts for fuzzy topological spaces. Besides some existing concepts of fuzzy R0, R1, T0, T1, T2 and regular topological spaces are recalled, and studied in detail. Interrelations among various separation axioms of fuzzy topological spaces are discussed. This book will be helpful for the undergraduate and graduate students, particularity for those who are studying Fuzzy Mathematics.

## Recent Advance Topics in Fuzzy Topological Spaces Using Mathematics”

Achievements And Awards In Abroad: YOUNG SCIENTIST (DST). MALAYASIA, 2010-2011. 1. International Conference On Mathematics And Statistics Held At Kuala-Lampur, Malayasia In 3 - 5, November 2010 - 2011. Selected As Young Scientist (DST), SENIOR SCIENTIST (DST). LONDON, U.K. 2011-2012. 2. World Science Congress (LONDON) Is An International Conference Of Applied And Engineering Mathematics, Selected As A Senior Scientist Held At London, U.K. In July 6 - 9, 2011 – 2012. (DST). KEY FEATURES: Each topic is treated in systematic, logical and lucid manner. The Scope and depth of coverage of each topic are strictly according to the world wide fuzzy topological space syllabus. Concepts are introduced in a sequential way with detailed explanations and illustrations. This book is entirely useful in fuzzy logic Programme and Fuzzy Topological Spaces.

## Fixed Point Theorems in Topological Spaces with Application to Fratal

The problems associated in the present research work which are taken care of are related to the study of fixed point theorems in semi Hausdorff space, uniform space and Banach space along with studying the applications of fixed point theorem in economics and analyzing the theory of fractals and application of fixed point theory. In this work, the introductory part of fractal theory, the important applications of fractal in image compression and a model of fixed point theorem in dynamical system is taken into consideration keeping in view its application in various areas like planetary motion, stock market prices, meteorological data etc. Also theorem of maximal elements in Banach space and existence of maximal elements and equilibrium for pair of maps in

## World Mixed Series in Bi-Topological Spaces : Using Mathematics

Each topic is treated in systematic logical and lucid manner. The Scope and depth of coverage of each topic are strictly according to the world wide fuzzy topological space syllabus.Concepts are introduced in a sequential way with detailed explanations. Dr. Sadanand N. Patil is a working as Assistant Professor and H.O.D in the D/o. Mathematics, KLS’s, VDRIT, HALIYAL. Karnataka, INDIA. Affiliated to VTU, BELGAUM. Since last 5 yrs he is teaching to UG & PG students of engineering stream. He has obtained M.Sc. (Mathematics) in the year 2002, PGDCA, in the year 2003 and He has obtained Ph.D. in Mathematics in the year 2009. Under the K U, Dharwad. The title of the thesis is SOME RECENT ADVANCE TOPICS IN TOPOLOGY, (FUZZY TOPOLOGICAL SPACES). Presently he is guiding to 6 candidates for Ph.D. in the field of FUZZY LOGIC Using FUZZY TOPOLOGICAL SPACES. As young scientist (DST) presented 2 papers in International Conference On Mathematics And Statistics Held At Kuala-Lampur, Malayasia (3 -5, November 2010). As senior scientist (DST) presented 2 papers in World Science Congress (LONDON) Is An IAENG , London, UK (July 6 - 9, 2011.) He Presented 5 Papers International Conference in Abroad, Presented 42 Papers In Various National and International Conferences and also Published 40 Papers In Various International Reputed Journals.

## Fuzzy Probability Spaces, Fuzzy Dynamical Systems, and Entropy

When planning experiments it is very important to know the amount of information that we obtain from their realization. It is known that a measure of information is entropy. A usual mathematical model of experiment in the classical information theory is a measurable partition. Partitions are standardly defined in the context of classical sets. It has appeared however, that for a solution of real problems are partitions defined by means of the concept of fuzzy sets more appropriate. The book contains a review of results of author related to the study of entropy in the fuzzy case. These results represent fuzzy generalizations of some concepts from the classical probability theory. The entropy of fuzzy partition can be considered as a measure of information of experiment, the results of which are fuzzy events. The concept of entropy of a fuzzy partition is exploited to define the entropy of fuzzy dynamical systems. Subsequently an ergodic theory for these systems is proposed. Presented results include the corresponding results from the classical ergodic theory as particular cases. The publication might be useful for specialists who deal in their research work with similar issues.

## Introduction to Fine Topological Spaces

This book describes a new concept of fine topological space. The collection of fine open sets contains all semi-open, pre-open, ?-open,?-open etc. sets which are used in defining the lighter concepts of continuity. With this wider class of fine sets the authors have defined the continuity which includes several other continuities already defined. Csaszar A. has introduced the concept of generalized topological space in 2002. A fine topological space is a special case of generalized topological space, but it may be noted that the concept of fine space is based on a topological space. The notions of several continuous functions in topological spaces and fuzzy topological spaces are widely developed which are used extensively in many practical and engineering problems. The concept of continuity and homeomorphism are used in many fields such as in Chemistry, Computer Science, Physics, Zoology, Quantum physics, Quantum mechanics etc. Now, there is a wide scope of study this concept in view of fuzzy topological space.

## On Lacunary Ideal Convergence in Intuitionistic Fuzzy n-Normed Spaces

This book deals with a concise study of convergence in intuitionistic fuzzy n-normed linear spaces. This book mainly contains the author's own research work in the area of lacunary ideal convergence. Fuzzy normed spaces have been an increasingly popular area of mathematical research in recent times, both in terms of theory and applications. But the availability of books in the area of fuzzy normed spaces is very rare. This book provides a good discussion on the development of both fuzzy and intuitionistic fuzzy set theory. The transition from fuzzy normed linear spaces to intuitionistic fuzzy n-normed linear spaces has been presented systematically. Anybody interested in the theory or application of fuzzy or intuitionistic fuzzy normed spaces will find this book more than useful. The book is written in such a way that mathematical prerequisites are minimum. Since the main subject of study in this book is a generalisaton of the concept of usual convergence, so all the related results in convergence have been incorporated in the book. This book may be used as a ready reference for an up to date account of results in the theory of fuzzy/intuitionistic fuzzy normed linear spaces.

## On Double Topological Structures

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, continuity, proximity, uniformity, and syntopogenous structures. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology. The notions of fuzzy sets and intuitionistic fuzzy sets used to generalize the topological spaces. Recently, Garcia and Rodabaugh could end some doubts around the term “intuitionistic” by giving it the new name “double”. Since that time, the new term is used to study the topological notions.

## Three papers on operator algebras in geometric topology

This short monograph offers a new vista of geometric topology from the standpoint of operator algebras. The common topic are functors from topological and geometric spaces to a category of operator algebras; such functors help to solve open problems in topology which are out of reach otherwise. Chapter 1 deals with new invariants of three-dimensional manifolds obtained from operator algebras. Chapter 2 links homology invariants of three-dimensional manifolds with the K-theory of Cuntz-Krieger algebras. Chapter 3 settles the Harvey conjecture for the mapping class group using ideas and methods of the noncommutative geometry.

## Lattice Valued Intuitionistic Fuzzy topological Structures

In this book, we study the lattice valued intuitionistic fuzzy topological spaces, lattice valued intuitionistic fuzzy syntopogenous structures, lattice valued intuitionistic proximity and lattice valued intuitionistic fuzzy uniformity in Entourag approach. Also, we construct stratified intuitionistic L-topological structures from a given intuitionistic L-topological structures. Finally, we introduce a new intuitionistic LC-uniformity in term of covering approach

## Characterization of 0-1 Possibility Measure

The purpose of this work is to investigate the topological nature of Fuzzy measure and relationship on the continuities between possibility density function and Possibility measure on compact topological spaces. The research deals with the concept of compactness of fuzzy topological spaces, continuities and regularity between Possibility measure and possibility density function of compact topological spaces and related ideas.

## Topological Quantum Field Theory in Two Dimensions

In my book, I give the analogy of the main results in one of Kevin Costello’s paper for open-closed topological conformal field theory. In other words, I show that there is a Batalin-Vilkovisky algebraic structure on the open-closed moduli space (moduli space of Riemann surface with boundary and marked points) , which is defined by Harrelson, Voronov and Zuniga, and the most important, there is a solution up to homotopy to the quantum master equation of that BV algebra if the initial condition is given, under the assumption that a new geometric chain theory gives rise to ordinary homology. This solution is expected to encode the fundamental chain of compactified open-closed moduli space, which is studied thoroughly by C.-C.Liu, as exactly in the closed case. We hope this result can give new insights into the mysterious two dimensional open-closed field theory.

## Rough & Fuzzy Computing Models: An Algebraic and Topological Approach

It has been shown that the algebraic and topological study of automata theory played a key role in the development of fundamentals of computer science. This book demonstrates some algebraic and topological aspects of rough automata and fuzzy automata. Specifically, this text present the foundations of algebraic theory of rough automata which will attract the researchers and will help in finding some successful applications of rough automata theory. Ideas for future research and explorations are also provided. Students and researchers in computer science and mathematics will benefit from this work.

## Fuzzy B-double star Opensets and fuzzy chi-double star Closed Sets

The concept of fuzzy set was introduced by Zadeh in his classical paper (Zadeh, 1965). Balasubramaniam and Sundaram defined generalized fuzzy closed sets in a fuzzy topological space and introduced certain types of fuzzy continuous functions between fuzzy topological spaces.(X, T) and (Y, S) denotes a fuzzy topological spaces.Also in this book I introduced different types of open sets and closed sets in fuzzy topological spaces.I think it will be very useful to co higher research in this field.