Elementary Topology
By O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, and V. M. Kharlamov
* Publisher: American Mathematical Society
* Number Of Pages: 400
* Publication Date: 2008-09-17
* ISBN-10 / ASIN: 0821845063
* ISBN-13 / EAN: 9780821845066
Product Description:
This textbook on elementary topology contains a detailed introduction to general topology and an introduction to algebraic topology via its most classical and elementary segment centered at the notions of fundamental group and covering space. The book is tailored for the reader who is determined to work actively. The proofs of theorems are separated from their formulations and are gathered at the end of each chapter. This makes the book look like a pure problem book and encourages the reader to think through each formulation. A reader who prefers a more traditional style can either find the proofs at the end of the chapter or skip them altogether. This style also caters to the expert who needs a handbook and prefers formulations not overshadowed by proofs. Most of the proofs are simple and easy to discover. The book can be useful and enjoyable for readers with quite different backgrounds and interests. The text is structured in such a way that it is easy to determine what to expect from each piece and how to use it. There is core material, which makes up a relatively small part of the book. The core material is interspersed with examples, illustrative and training problems, and relevant discussions. The reader who has mastered the core material acquires a strong background in elementary topology and will feel at home in the environment of abstract mathematics. With almost no prerequisites (except real numbers), the book can serve as a text for a course on general and beginning algebraic topology.
Elementary means close to elements, basics. It is impossible to determine
precisely, once and for all, which topology is elementary, and which
is not. The elementary part of a subject is the part with which an expert
starts to teach a novice.
We suppose that our student is ready to study topology. So, we do not
try to win her or his attention and benevolence by hasty and obscure stories
about misterious and attractive things such as the Klein bottle.1 All in good
time: the Klein bottle will appear in its turn. However, we start with what
a topological space is. That is, we start with general topology.
General topology became a part of the general mathematical language
long ago. It teaches one to speak clearly and precisely about things related
to the idea of continuity. It is needed not only in order to explain what,
finally, the Klein bottle is. This is also a way to introduce geometrical images
into any area of mathematics, no matter how far from geometry the area
may be at first glance.
As an active research area, general topology is practically completed. A
permanent usage in the capacity of a general mathematical language has
polished its system of definitions and theorems. Nowadays, study of general
topology indeed resembles rather a study of a language than a study of
mathematics: one has to learn many new words, while the proofs of the
majority of theorems are extremely simple. But the quantity of the theorems
is huge. This comes as no surprise because they play the role of rules that
regulate usage of words.
The book consists of two parts. General topology is the subject of the
first one. The second part is an introduction to algebraic topology via its
most classical and elementary segment which emerges from the notions of
fundamental group and covering space.
In our opinion, elementary topology also includes basic topology of manifolds,
i.e., spaces that look locally as the Euclidean space. One- and twodimensional
manifolds, i.e., curves and surfaces, are especially elementary.
But a book should not be too thick, and so we had to stop.
Chapter 5 keeps somewhat aloof. Its material is intimately related to a
number of different areas of Mathematics. Although it plays a profound role
in these areas, it is not that important in the initial study of general topology.
Therefore mastering of this material may be postponed until it appears in a
substantial way in other mathematical courses (which will concern the Lie
groups, functional analysis, etc.). The main reason why we included this
material is that it provides a great variety of examples and excercises
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