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Review Essays of Academic, Professional & Technical Books in the Humanities & Sciences


Differential Geometry and Topology: With a View to Dynamical Systems by Keith Burns, Marian Gidea (Studies in Advanced Mathematics: Chapman & Hall/CRC)
Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology; and dynamical systems. Topics of special interest addressed in the book include Brouwer's fixed point theorem, Morse Theory, and the geodesic flow.

Smooth manifolds, Riemannian metrics, affine connections; the curvature tensor, differential forms, and integration on manifolds provide the foundation for many applications in dynamical systems and mechanics. The authors also discuss the Gauss-Bonnet theorem and its implications in non-Euclidean geometry models.

The differential topology aspect of the book centers on classical, transversality theory, Sard's theorem, intersection theory, and fixed-point theorems. The construction of the de Rham cohomology builds further arguments for the strong connection between the differential structure and the topological structure. It also furnishes some of the tools necessary for a complete understanding of the Morse theory. These discussions are followed by an introduction to the theory of hyperbolic systems, with emphasis on the quintessential role of the geodesic flow.

The integration of geometric theory, topological theory, and concrete applications to dynamical systems set this book apart. With clean, clear prose and effective examples; the authors' intuitive approach creates a treatment that is comprehensible to relative beginners, yet rigorous enough for those with more background and experience in the field.

Marian Gidea is an associate professor in the Department of Mathematics, Northeastern Illinois University, Chicago, USA.

Keith Burns is a professor in the Department of Mathematics, Northwestern University, Evanston, Illinois, USA.

General Topology by Stephen Willard (Dover Publications, Inc.) Among the best available reference introductions to topology, General Topology is appropriate for advanced undergraduate and beginning graduate students. Written by Stephen Willard, University of Alberta, this is a Dover unabridged republication of the edition first published by Addison-Wesley Publishing Company, Reading, Massachusetts, 1970.

The volume gives balanced treatment to two broad areas of general topology, continuous topology and geometric topology. The first, continuous topology, centers on the effects of compactness and metrization, is represented here by sections on convergence, compactness, metrization and complete metric spaces, uniform spaces, and function spaces. The second, geometric topology, focuses on the connectivity properties of topological spaces and provides the core results from general topology that serve as background for subsequent courses in geometry and algebraic topology. This core is formed here by a series of nine sections on connectivity properties, topological characterization theorems, and homotopy theory.

The chapters are divided into sub-topics that progress from introductory notes on essential set theory through subspace, products, compactness, separation and countability axioms, compactifications, and function spaces. Many of general topology's standard spaces are introduced and examined in the generous number of related problems that accompany each section – 340 in all.

General Topology’s value as a reference work is enhanced by a collection of historical notes for each section, an extensive bibliography, and an index. The book is intended as both a text and reference and is paced slowly at the beginning to be an aid to students new to topology.

Fuzzy Topology by N Palaniappan (CRC Press) imparts the latest developments for graduate students and research scholars covering various properties of fuzzy topology viz., fuzzy point and its neighborhood structure, fuzzy nets and fuzzy convergence, fuzzy metric spaces, different fuzzy compactness, fuzzy connectedness, fuzzy separation axioms and properties, product spaces, convex fuzzy sets and fuzzy uniform spaces.

The key features are

  • Large number of examples
  • Counterexamples, characterizations, implications
  • References to original sources

In recent years, many concepts in mathematics, engineering, computer science, and many other disciplines have been in a sense redefined to incorporate the notion of fuzziness. Designed for graduate students and research scholars, Fuzzy Topology imparts the concepts and recent developments related to the various properties of fuzzy topology.

The author first addresses fundamental problems, such as the idea of a fuzzy point and its neighborhood structure and the theory of convergence. He then studies the connection between fuzzy topological spaces and topological spaces and introduces fuzzy continuity and product induced spaces. Chapter Three examines fuzzy nets, fuzzy upper and lower limits, and fuzzy convergence and is followed by a study of fuzzy metric spaces. The treatment then introduces the concept of fuzzy compactness before moving to initial and final topologies and the fuzzy Tychnoff theorem. The final sections of the book cover connectedness, complements, separation axioms, and uniform spaces.  

Only in twentieth century, mathematicians defined the concepts of sets and functions to represent problems. This way of representing problems is more rigid. In many circumstances the solutions using this concept are meaningless. This difficulty was overcome by the fuzzy concept. Almost all mathematical, engineering, medicine, etc. concepts have been redefined using fuzzy sets. Hence it is a must to popularize these ideas for our future generation. This makes me to make an attempt in bringing out this book.

In order to study the control problems of complicate systems and dealing with fuzzy informations, American Cyberneticist L. A. Zadeh introduced Fuzzy Set Theory in 1965, describing fuzziness mathematically for the first time. Following the study on certainty and on randomness the study of mathematics began to explore the previously restricted zone‑fuzziness. Fuzziness is a kind of uncertainty. Since the 16th century, probability theory has been studying a kind of uncertainty-randomness, i.e., the uncertainty of the occur of an event: but in this case, the event itself is completely certain, the only uncertain thing is whether the event will occur or not, the casuality is not completely clear now. However, there exist another kind of uncertainty‑fuzziness, i.e. for some events, it can not be completely determined that which cases these events should be subordinated to (e.g., they have already occurred or have not occurred yet), they are in a nonblack and nonwhite state‑that is to say, the law of excluded middle in logic can not be applied any more. Which case an event should be subordinated to, in mathematical view, is just that which set the "element" standing for the event should belong to. However, in Mathematics, a set A can be equivalently represented by its characteristic function ‑ a mapping XA from the universe X of discourse (region of consideration, i.e., a larger set) containing A to the 2‑value set [0,1): i.e. , it is to say, x belongs to A if and only if XA(x) = 1. But in"fuzzy" case "belonging to" relation XA(x) between xandA is no longer "0 or otherwise 1," it has a degree of "belonging to," i.e., membership degree, such as 0.6. Therefore, the range has to be extended from (0,1) to [0,1]; or more generally, a lattice L, because all the membership degrees, in mathematical view, form an ordered structure, a lattice. A mapping from X to [0,1] or a lattice L called a generalized characteristic function describes the fuzziness of "set" in general. A fuzzy set on a universe X is simply just a mapping from X to [0,1] or to a lattice L.

Thus, fuzzy set extended the basic mathematical concept-set. In view of the fact that set theory is the cornerstone of modern mathematics, a new and more general framework of mathematics was established. Fuzzy mathematics is just a kind of mathematics developed in this framework, and fuzzy topology is just a kind of topology developed on fuzzy sets. Hence, fuzzy mathematics is a kind of Mathematical Theory which contains wider content than the Classical Theory.

Denote the family of all the fuzzy sets on the universe X, which takes [0,1] as the range, by IX, where I = [0,1 ]. Substituting inclusive relation by the order relation in IX, we introduce a topological structure naturally into IX. So that fuzzy topology is a common carrier of ordered structure and topological structure. According to the point of view of Bourbakian School , there are mainly three large kinds of structures in Mathematics‑topological structure, algebraic structure, and ordered structure. Fuzzy topology fuses just two large structures‑ ordered structure and topological structure. Therefore, even if we consider only its pure mathematical significance but not its practical background, fuzzy topology do has important value to research. Fuzzy topology naturally possesses "pointlike" structure. This structure is a basic characteristic in fuzzy topology. To illustrate this point, we can take the problem of "membership relation" between a point and a set in fuzzy topology as an example. In classical topology, this relation is simple and clear: "An open set is a neighborhood of a point if and only if this point belongs to this open set." In early period of fuzzy topology, "membership relation" was similarly defined. But serious difference, soon surfaced from this basic definition in fuzzy topology. For example, under this definition, one point can belong to a union of some sets but does not belong to any one of them. This caused many other relative problems. In otherwords the so‑called "multiple choice principle" no longer holds. This obstacle was overcome in 1977. In this year, theory of quasi‑coincident neighborhood system introduced by Liu Ying‑Ming made break through in this respect.

As is well known, neighborhood structure can be decomposed as:

"Structure of open sets + Membership Relation between Point and Set." The "membership relation" corresponding to traditional neighborhood system is just "relation of belonging to." For making the membership relation between point and set in fuzzy topology satisfying the very basic "multiple choice principle" mentioned above and some other obvious requirements, Liu proved that in the frame work of fuzzy sets this membership relation can only be the so‑called "quasicoincidence relation", but in general not "relation of belonging to". The neighborhood structure corresponding to the quasi‑coincidence relation is just the quasi‑coincident neighborhood system. In this theory, a "point" can be in the outside of its neighborhood structure‑ quasi‑coincident neighborhood. This kind of topology construction, in which points do not "belong to" their neighborhood structure, was investigated early in 1916 by French Mathematician Freche't. The research was summarized latter as "V‑Space Theory" in Sierpinski's monograph General Topology. However, in V‑Space Theory, the intersection of a set and its complement is always empty. Since the law of excluded middle is no longer valid in fuzzy sets its property no longer holds either. Therefore, fuzzy topology and V‑Space Theory are two kinds of difference theories. This example also shows that the study of fuzzy topology can deepen our understanding of some most basic structure (e.g., neighborhood structure) in classical mathematics.

In fact, point like structure of g fuzzy set is a kind of behaviour of their level structures, or in other words, stratifications. For every fuzzy set A: X ‑)‑ I and every element a E 1, the set Alai ~= ]x E X A(x) >_ a] is a a‑level of possessing level structures and is the most essential characteristic of fuzzy sets. Obviously, A can be formed from its stratifications. Thus, the relation between fuzzy sets and ordinary sets (stratifications) is established. May be the converse problem is more interesting: For a certain family of ordinary sets, how to construct a fuzzy set (a mapping) from the family with some desired properties?

To show the effects of stratifications and the associate methods, let us observe

the following classical Hahn‑Diendonne'‑ Tong insertion theorem. Let X be a topological space, f, g: X‑‑> [0,1] upper semi‑continuous function and lower semi continuous function from X to unit interval, respectively, f :S g, then X is normal if and only if there exists a continuous function h: X‑> [0, 1] such that f < h _< g. The proof of this theorem, in otherwords, the determination of the inserting function h, is pointwisely obtained, full of analytic techniques, its arguments considerably complicated. Based on the understanding of set theoretical relations and the topological relations among a mapping and its stratifications, the stratification method can be used to construct mappings level by level, which successfully solved this inserting mapping problem and generalized this noted

Theorem in classical mathematics. Compared with the original analytic Techniques, this proof based on tht; stratification method is more effective, and is natural, simple and conceptual. This example also shows that the study of fuzzy can offer us new methods and stronger conclusions. Although the peculiar level structures of fuzzy topological spaces makes some problems complicated, however, it is just level structure itself which makes fuzzy topological spaces possesses more abundant properties, making the relation between fuzzy topology and other branches of classical mathematics closer.

Fuzzy is a generalization of fuzzy topology in classical mathematics, but it also

has its own marked characteristics. Also, it can deepen the understanding of basic structure of classical mathematics, offer new methods and results, and obtain significant results of classical mathematics. Moreover, it also has applications in some important aspects of science and technology.

The first chapter concerns to fundamental problems. The first one is the concept of a fuzzy point and its neighborhood structure. The second problem concerns the theory of convergence. In the second chapter two functors W and i are introduced to study connection between fuzzy topological spaces and topological spaces. Also fuzzy continuity and product induced spaces have been introduced. Fuzzy nets, fuzzy upper and lower limit, fuzzy convergence are dealt in Chapter 3. Chapter 4 is the study of fuzzy metric spaces. The concept of various fuzzy compactness have been introduced in Chapter 5. Initial and final topologies and the fuzzy Tychnoff Theorem is studied in Chapter 6. The remaining chapters cover connectedness, complements, separation axioms and uniform spaces.

Different notations have been used in different places for the fuzzy point, fuzzy metric, fuzzy topology, etc for convenience. The readers are advised to go through Chapter 8 after Chapter 3 for clear understanding of remaining chapters.


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