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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Often algebraic topology texts assume that the reader is well acquainted with arguments of a previous course in point-set topology like this in order not to get trapped on details. East Dane Designer Men’s Fashion.
Amazon Restaurants Food delivery from local restaurants. Also, Lagrange’s theorem is also immediate in your sense because quotienting a finite group by a subgroup is precisely the collapsing of its cosets to a point, so clearly the number of elements in hanich quotient is exactly the number of cosets.
I believe that experienced mathematicians, who perhaps learned 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 topolkgy I have seen.
I might be misinterpreting you, but I think that you are saying that you do not like my suggestion that cell-complex quotients are immediately viewed as a wedge otpology of n-spheres. Set up a giveaway. Discover Prime Book Box for Kids. The author then delves into topological groups and vector spaces introducting Hilbert Banach and Frechet spaces albeit briefly. I especially enjoyed the section on covering spaces with which it concludes.
algebraic topology – How much rigour is necessary? – Mathematics Stack Exchange
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The German mathematician Klaus Janich has a wonderful response to this question in his book on topology, which is intentionally very non-rigorous and intuitive:. It is often said against intuitive, spatial argumentation that it jancih not really argumentation,but just so much gesticulation-just ‘handwaving’.
Sometimes in mathematics it is difficult to see the forest for the trees. Dignaga 2 5. There was a problem filtering reviews right now. 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. Most of the jwnich until the last chapters are of the 1- or 2-paragraph variety, with some pictures added, although as the book progresses the level becomes increasingly more sophisticated.
Quotient spaces,homotopy, complexes and urysohn and tietze lemma along with partitions of unity are tackled next. Topology Undergraduate Texts in Mathematics by K. For me, this level of “rigor” required lies somewhere between explicitly writing out everything in bare bones set theoretic terms, and the level of detail presented in a graduate analysis text such as Rudin.
But since algebraic topology is so closely related to classical geometry, completely abstract reasoning would probably strip away much understanding of the sources of most of the central concepts,which I believe was Hatcher’s reason for writing the text in this manner. So allow me to clarify. 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.
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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. See all 6 reviews. You will take away so much for the price paid. Well said, although the thing I really don’t like about intuitive handwaving is that intuition differs from person to janicb. The question of what constitutes a proof has constantly been questioned and revised since the beginnings of mathematics jsnich the Ancient World.
How much rigour is necessary?
Topology : Undergraduate Texts in Mathematics
The reader jnaich walk away with an appreciation of the idea that topology is just not abstract formalism, but has an underlying janicy that is rich in imagery. Showing of 6 reviews. There’s a problem loading this menu right now. The book also covers both point-set topology topological spaces, compactness, connectedness, separation axioms, completeness, metric topology, TVS, quotient topology, countability, metrization, etc.
Whenever I feel that I am just grinding away at technical results, this is one of the books I return to for a little perspective.