With a nice balance of mathematical precision and accessibility, this text provides a broad introduction to the field of topology. Author Sue Goodman piques student curiosity and interest without losing necessary rigor so that they can appreciate the beauty and fun of mathematics. The text demonstrates that mathematics is an active and ever-changing field with many problems still unsolved, and students will see how the various areas of mathematics – algebra, combinatorics, geometry, calculus, and differential equations – interact with topology. Students learn some of the major ideas and results in the field, do explorations and fairly elementary proofs, and become aware of some recent questions. By presenting a wide range of topics, exercises, and examples, Goodman creates an interactive and enjoyable atmosphere in which to learn topology.
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THE EULER CHARACTERISTIC
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boundary circles bracket polynomial called Cantor set cell complex cell decomposition Chapter classification theorem colors compact complete graph compute connected sum connectedness COROLLARY critical points crossing number defined deformation retract disk Draw embed embedded Euler characteristic example EXERCISE finite number fractal fundamental group genus gives gluing instructions gradient vector field Hausdorff Hence homeomorphic identified integer integral curves interval invariant isomorphic Klein bottle knot diagram lemma limit point minus Mobius band neighborhood nonorientable NOTE Notice number of crossings number of edges open set orbit orientable surface path connected path homotopy phase portrait projective plane proof PROPOSITION prove regular complex Reidemeister moves result Section Seifert circles Seifert surface shown in Figure simple closed curve simply connected space model spanning surface sphere square standard plane model subset Suppose surface with boundary tangent topological property topological space torus triangle unknot vector field vertex vertices words