## Beginning TopologyBeginning Topology is designed to give undergraduate students a broad notion of the scope of topology in areas of point-set, geometric, combinatorial, differential, and algebraic topology, including an introduction to knot theory. A primary goal is to expose students to some recent research and to get them actively involved in learning. Exercises and open-ended projects are placed throughout the text, making it adaptable to seminar-style classes. The book starts with a chapter introducing the basic concepts of point-set topology, with examples chosen to captivate students' imaginations while illustrating the need for rigor. Most of the material in this and the next two chapters is essential for the remainder of the book. One can then choose from chapters on map coloring, vector fields on surfaces, the fundamental group, and knot theory. A solid foundation in calculus is necessary, with some differential equations and basic group theory helpful in a couple of chapters. Topics are chosen to appeal to a wide variety of students: primarily upper-level math majors, but also a few freshmen and sophomores as well as graduate students from physics, economics, and computer science. All students will benefit from seeing the interaction of topology with other fields of mathematics and science; some will be motivated to continue with a more in-depth, rigorous study of topology. |

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boundary circles bracket polynomial called Cantor set cell complex cell decomposition Chapter classiﬁcation theorem compact complete graph compute connected sum connectedness COROLLARY critical points crossing number deﬁned deﬁnition deformation retract difﬁcult dimension disk Draw embed embedded Euler characteristic example EXERCISE ﬁgure ﬁnd ﬁnding ﬁnite number ﬁrst ﬁxed fractal fundamental group genus gives gluing instructions gradient vector ﬁeld 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 projective plane proof PROPOSITION prove regular complexes Reidemeister moves result Section Seifert circles Seifert surface shown in Figure simple closed curve space model spanning surface square standard plane model subset Suppose surface with boundary tangent topological property topological space torus triangle unknot unknotting number vertex vertices words