Lecture Notes on Elementary Topology and Geometry

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Springer Science & Business Media, Dec 10, 1976 - Mathematics - 232 pages
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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Contents

II
V
3
VI
9
VII
11
VIII
14
IX
18
X
24
XIII
29
XXV
88
XXVI
92
XXVII
107
XXX
116
XXXI
130
XXXII
151
XXXV
159
XXXVI
173

XIV
32
XV
38
XVI
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XVII
47
XX
50
XXI
60
XXII
76
XXIII
77
XXIV
81
XXXIX
182
XL
188
XLI
196
XLII
205
XLIII
214
XLV
228
XLVI
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