Algebraic Topology: An Intuitive Approach (Translations of by Hajime Sato

By Hajime Sato

The one so much tough factor one faces while one starts to benefit a brand new department of arithmetic is to get a consider for the mathematical feel of the topic. the aim of this ebook is to assist the aspiring reader collect this crucial good judgment approximately algebraic topology in a quick time period. To this finish, Sato leads the reader via basic yet significant examples in concrete phrases. additionally, effects are usually not mentioned of their maximum attainable generality, yet when it comes to the easiest and so much crucial situations. based on feedback from readers of the unique version of this e-book, Sato has extra an appendix of valuable definitions and effects on units, common topology, teams and such. He has additionally supplied references.Topics coated contain primary notions equivalent to homeomorphisms, homotopy equivalence, basic teams and better homotopy teams, homology and cohomology, fiber bundles, spectral sequences and attribute periods. items and examples thought of within the textual content contain the torus, the Mobius strip, the Klein bottle, closed surfaces, mobile complexes and vector bundles.

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Extra resources for Algebraic Topology: An Intuitive Approach (Translations of Mathematical Monographs, Volume 183)

Example text

Interchanging the roles of x and y, one sees that d(x, A) - d(y, A) s d(x, y) + s. Therefore Id(x, A) - dey, A) I s d(x, y) + s. This being true for every s > 0, one obtains Id(x, A) - d(y, A) I s d(x, y). 2. 1. Theorem. Let E be a metric space, AcE, x conditions are equivalent: ~ E E. The following (i) xEA; (ii) there exists a sequence (Xl' x 2 , ... fpoints of A that tends to x. ' therefore intersects A; consequently x EA. If x E A then, for every integer n ~ 1, there exists a point x. of A that belongs to the closed ball with center x and radius lin.

Since U 2 may be identified with the surface of the space commonly called a 'torus', one says that T2 is the 2-dimensional torus, and more generally that Tn is the n-dimensional torus. In particular, T is called the I-dimensional torus . 17. Theorem. finite number of compact spaces is compact. It suffices to show that if X and Yare compact, then X x Y is compact. 5). Let (Uj);EI be an open covering of X x Y. 4, there exist an open neighborhood Vm of x in X and an open neighborhood W m of y in Y such that Vm x Wm c U;(m)' Set Pm = Vm x W m .

15. Corollary. Let E be a compact space, F a separated space, f a continuous bUective mapping ofE onto F. Then f - 1 is continuous (in other words, f is a homeomorphism of E onto F). Let 9 = f - I. 7), in other words g-'(A) is closed in F. 4). 16. Example. Let p be the canonical mapping of R onto T. 2). 12). But p([O, I]) = T. Thus the space T is compact. 5 we defined a continuous bijectionf ofT onto U. Now, T is compact and U is separated. 15). Thus, the spaces T and U are homeomorphic. Since U 2 may be identified with the surface of the space commonly called a 'torus', one says that T2 is the 2-dimensional torus, and more generally that Tn is the n-dimensional torus.

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