First Concepts of Topology: The Geometry of Mappings of Segments, Curves, Circles, and Disks
The authors of First Concepts of Topology demonstrate the power, the flavor and the adaptability of topology, one of the youngest branches of mathematics, in proving so-called existence theorems. An existence theorem asserts that a solution to some given problem exists; thus it assures those who hunt for a solution that their labors may not be in vain. Since existence theorems are frequently basic to the structure of a mathematical subject, the applications of topology to the proofs of these theorems are frequently basic to the structure of a mathematical subject, the applications of topology to the proofs of these theorems constitute a unifying force for large areas of mathematics. In Part I of this monograph an existence theorem governing a large class of one-dimensional problems is treated; all the important ingredients in its proof, such as continuity of functions, compactness and connectedness of point sets, are developed and illustrated. In Part II, its two-dimensional analogue is carefully built via the necessary generalizations of the one-dimensional tools and concepts. The results are applied to such fundamental mathematical objects as zeros of polynomials, fixed points of mappings, and singularities of vector fields. The reader will find that each of the new concepts he masters will prove to be of invaluable help in his mathematical progress, especially in understanding the basis of the calculus. -- from back cover.
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First concepts of topology: the geometry of mappings of segments, curves ...
William G. Chinn,Norman Earl Steenrod
Affichage d'extraits - 1966
A U B angle antipodes bounded set circle closed and bounded closed curve closed interval closed set compact complement completes the proof complex numbers concept conﬁguration connected set constant continuous function contracting sequence coordinates decimal expansion deﬁned deﬁnition deleted denote diameter diametrical points disk distance divide endpoint equation example Exercises exterior f is continuous Figure ﬁnd ﬁne ﬁnite covering ﬁnite number ﬁrst ﬁxed point ﬂow follows function f geometry graph hence homeomorphism homotopy inﬁnite integer interior intersection inverse image leaves ﬁxed length Let f line segment main theorem mapping f mathematics neighborhood obtain open covering open interval open set pair parallel partition perpendicular projection plane polynomial protractor prove radial projection radius rational numbers real numbers rectangle reﬂection rotation Section single point solution sphere subset suﬂiciently tangent topological property topologically equivalent union vector ﬁeld velocity winding number y-value