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K. Jänich, "Topology (Undergraduate Texts in Mathematics)"

Contents: Introduction. - Fundamental Concepts. - Topological Vector Spaces.- The Quotient Topology. - Completion of Metric Spaces. - Homotopy. - The Two Countability Axioms. - CW-Complexes. - Construction of Continuous Functions on Topological Spaces. - Covering Spaces. - The Theorem of Tychonoff. - Set Theory (by T. Br|cker). - References. - Table of Symbols. -Index.

Reviews:

Summary: A simple introduction to advanced mathematical concepts

This text gives the reason behind many advanced topological concepts and tantalizes the reader with it’s varied applications.

Basic topological concepts of open, closed, continuous, product topology, connectedness,compactness and intro to separation axioms is presented in a logical concise and easy to understand way.

The author then delves into topological groups and vector spaces introducting Hilbert Banach and Frechet spaces ( albeit briefly ).

Quotient spaces,homotopy, complexes and urysohn and tietze lemma along with partitions of unity are tackled next.

I especially enjoyed the section on covering spaces with which it concludes.

Perhaps the single best accolade I can give the book is that it gives one inspiration and motivation to explore in greater detail mathematical objects discussed.

The text is useful to all students of mathematics and physics alike.

Summary: Students: BUY THIS BOOK!!!

Rating: 5

It is not too often that a book about topology is written with the goal of actually explaining in detail what is going on behind the formalism. The author does a brilliant job of teaching the reader the essential concepts of point set topology, and the book is very fun to read. The reader will walk away with an appreciation of the idea that topology is just not abstract formalism, but has an underlying intuition that is rich in imagery. The author has a knack for allowing readers to “see into the future” of what kind of mathematics is waiting for them and how topology is indispensable in its study.

At the end of chapter three, which deals with the quotient topology, the author writes the following paragraph: “If is often said against intuitive, spatial argumentation that it is not really argumentation but just so much gesticulation - just ‘handwaving’. Shall we then abandon all intuitive arguments? Certainly not. As long as it is backed by the gold standard of rigorous proofs, the paper money of gestures is an invaluable aid for quick communication and fast circulation of ideas. Long live handwaving!”. This has to rank as one of the best paragraphs that has every appeared in a mathematics book, for it nicely summarizes the need for developing a feel for the concepts behind mathematics before moving on to the rigorous proofs. Physicists in particular, who must assimilate mathematics very quickly in order to apply it to real problems must have a pictorial, “playful” understanding of the mathematical constructions.

Thus the language that the author employs is informal, and a listing of the best discussions in the book would really entail a listing of every one in the book. There is not one part of the book that is not helpful or interesting, and the author delves into many different areas that involve the use of topology.

If you are a beginning student in mathematics, BUY AND STUDY THIS BOOK…BUY AND STUDY THIS BOOK. You will take away so much for the price paid.

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Rating:
2.5 out of 5 by Book123