Download A Combinatorial Introduction to Topology (Dover Books on by Michael Henle PDF

By Michael Henle

First-class textual content for upper-level undergraduate and graduate scholars indicates how geometric and algebraic principles met and grew jointly into a massive 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 workouts (some suggestions) built-in into the textual content. 1979 version. Bibliography.

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Example text

Imagine each point of Q) labeled according to the direction of the vector field V. If @ contains complete triangles of arbitrarily small size, then there is a point P in Q) where V(P) = 0. By hypothesis, Q) contains a sequence of complete triangles with vertexes Pn, Qn, and Rn labeled A, B, and C, respectively, such that the length of the sides of these triangles tends to zero. Thus there are three sequences of vertexes 0> = {Pn}, 1 = {Q„}, and 0t = {Rn} consisting only of points with northeast-, northwest-, and south-pointing vectors, respectively.

Prove that the set S is open if and only if S" is closed. 16. For any set S the boundary of 5, b(S), consists of those points that are near both S and S'. Find the boundary of each set in Exercise 3. Show that b(S) is the intersection of the closures of S and S'. 17. Show that a set S is open if and only if S does not contain a single point of b(S). Show that S is closed if and only if S contains all of b(S). Use these results to give a second proof for Exercise 15. 18. For any set S prove that every point of the plane is in exactly one of the sets I(S\ I(S') and b{S).

To find a fixed point for g, consider the composition / = v o g o w, where i; is the inverse transformation v:R^> D. This composition is a continuous transformation of D to D. By hypothesis, / has a fixed point P in D, v(g(u{P))) = f(P) — P. This means that g(u(P)) = u(P\ so that u(P) is a fixed point for #. This proves that R has the fixed point property. The following theorem is the most important result on fixed points in the plane. Brouwer's Fixed Point Theorem Cells have the fixed point property.

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