TUTM. Undergraduate Texts in Mathematics. Klaus Jänich. This is an intellectually stimulating, informal presentation of those parts of point set topology that are. Topology by Klaus Janich: Forward. Content. Sample. Back cover. Review. This is an intellectually stimulating, informal presentation of those parts of point set topology that are of importance to the nonspecialist.
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This is the most pleasurable text on general topology you are likely to find. Just for the sections on Urysohn’s Lemma, Tietze’s Extension, and Tychonoff’s Theorem alone this book is worth owning – it contains the nicest proofs of these theorems I have ever seen.
The book is packed with revealing illustrations and motivations. Sometimes in mathematics it is difficult to see the forest for the trees. Topoligy I gopology that I am just grinding away at technical results, this is one of the books I return to for a little perspective. Topklogy I agree with the other reviewers here that Jaenich’s “Topology” is very well written, goes to great lengths to explain the “hows and whys” of topology, and includes many, many figures about 1 per page on averageit is probably more popular with people who already know topology than with beginning students, even though it is an introductory text intended for undergraduates.
This is due to both a frequent lack jqnich precision or formality janiich proofs and definitions coupled with a tendency to discuss much more advanced material with which a student at this level wouldn’t be familiar. I believe that experienced mathematicians, who perhaps topplogy point-set topology from books such as those of Munkres, Kelley, or Bredon or even an analysis book such as Roydenappreciate how this book focuses on motivating the concepts, explaining how the various objects are used elsewhere in mathematics – for that purpose this is one of the finest books I have seen.
However, too much material is mentioned that is certainly over the heads of most students new to topology, such as the Pontrjagin-Thom construction, the spectrum of commutative Banach algebras, or Lie groups, often in a very cursory manner that would serve only to confuse beginners. Concepts are often used before they janifh defined, or are not defined precisely, which is liable to frustrate these students as well.
Many topics are given such short attention it makes you wonder why the author even bothered – such as a page devoted to Frechet spaces followed by a section consisting of a single paragraph on locally convex topological vector spaces.
Much of the material is not covered very deeply – only a definition and maybe a theorem, which half the time isn’t even proved but just cited. Certainly this book couldn’t be used as a textbook for an undergraduate course – for the reasons mentioned above and also because not enough material is actually covered, as well as the tpology deficiency in that it lacks exercises for the reader. Most of the proofs until the last chapters are of the 1- or 2-paragraph janifh, with some pictures added, although as the book progresses the level becomes increasingly more sophisticated.
The book also covers both point-set topology topological spaces, compactness, connectedness, separation axioms, completeness, metric topology, TVS, quotient topology, countability, metrization, etc. Half of the chapter on homotopy is actually concerned with categories and functors, probably not the best way to introduce the subject. In fact, here is direct quote from the index: So why am I rating this 5 stars? For the wealth of examples e.
Just by looking at the contents one can see this, as there are sections titled: This is definitely the most fleshed-out part; if only the rest of the book could go into this depth. This book would make an excellent supplement to a more formal textbook such as Munkres, but is not a substitute for it.
But I would still consider this as a must-read for all those students who plan on studying tkpology in graduate school. 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: Shall we then abandon all intuitive arguments? 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.
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.
You will take away so much for the price paid. This text gives topologh 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. See all 6 reviews. Amazon Giveaway allows you to run promotional giveaways in order to create buzz, reward your audience, and attract new followers and customers.
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