Lecture Notes on Elementary Topology and Geometry
At the present time, the average undergraduate mathematics major finds mathematics heavily compartmentalized. After the calculus, he takes a course in analysis and a course in algebra. Depending upon his interests (or those of his department), he takes courses in special topics. Ifhe is exposed to topology, it is usually straightforward point set topology; if he is exposed to geom etry, it is usually classical differential geometry. The exciting revelations that there is some unity in mathematics, that fields overlap, that techniques of one field have applications in another, are denied the undergraduate. He must wait until he is well into graduate work to see interconnections, presumably because earlier he doesn't know enough. These notes are an attempt to break up this compartmentalization, at least in topology-geometry. What the student has learned in algebra and advanced calculus are used to prove some fairly deep results relating geometry, topol ogy, and group theory. (De Rham's theorem, the Gauss-Bonnet theorem for surfaces, the functorial relation of fundamental group to covering space, and surfaces of constant curvature as homogeneous spaces are the most note worthy examples.) In the first two chapters the bare essentials of elementary point set topology are set forth with some hint ofthe subject's application to functional analysis.
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arcwise connected assume ball basis called collection compact Condition connected consider constant construct containing continuous converges coordinate functions coordinate system Corollary covering space curvature curve defined Definition denote determined dimension disjoint domain element equal equation equivalence Example exists extended fact Figure finite function Furthermore geodesic given Hausdorff Hence homeomorphism homotopy identity implies induction integer interval isometry isomorphism lemma lift linear locally metric space Moreover neighborhood Note obtained open set ordered oriented origin parallel translation particular path projection Proof prove regular relative Remark Riemannian rotation route satisfied sequence Similarly simplex simplicial complex smooth smooth manifold structure subset suffices Suppose surjective tangent vector Theorem topological space topology union unique unit vector field vertex vertices zero
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Limited preview - 2006