Mathematics

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

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

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

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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