By Michael Henle

ISBN-10: 0486679667

ISBN-13: 9780486679662

First-class textual content for upper-level undergraduate and graduate scholars indicates how geometric and algebraic principles met and grew jointly into a big department of arithmetic. Lucid insurance of vector fields, surfaces, homology of complexes, even more. a few wisdom of differential equations and multivariate calculus required. Many difficulties and routines (some recommendations) built-in into the textual content. 1979 version. Bibliography.

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This has just one edge of A*-—-^£ positive orientation and so contributes one to S. At the same time as a complete triangle it counts one as part of C. As a second example consider B a triangle of type A\ . This has two edges AB, one each of positive A+-—-^ A and negative orientation, and so contributes a total of zero to S. It also contributes zero to C, since it is not complete. By surveying all the possible types of triangle, one verifies that each triangle is counted in the same way in S as in C; therefore C = S.

Compact)? } (h) the whole plane (i) the empty set 4. The property of being closed and bounded is a topological property, namely compactness. Show, however, by examples that neither closedness nor boundedness alone is a topological property. 5. Prove that Jordan curves are compact. 6. For any set 5, the closure of S consists of S plus all the points near S. Find the closure of each of the sets of Exercise 3. Prove that the closure of any set is closed. " The formal topological property corresponding to this intuitive phrase is called connectedness.

Since the sides of the triangles tend to zero, it follows that P is also near J and near ^ . A\ The vectors V(0>) all point northeast. Since the set of northeast-pointing vectors, namely the first quadrant, is closed, it follows that §7 PHASE PORTRAITS AND THE INDEX LEMMA 43 V(P) is northeast-pointing too. Similarly, V(P) is northwest-pointing and south-pointing. Since the zero vector is the only vector pointing all three directions, V(P) = 0, completing the proof of Brouwer's theorem and the lemma.

### A Combinatorial Introduction to Topology (Dover Books on Mathematics) by Michael Henle

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