First Concepts of Topology: The Geometry of Mappings of Segments, Curves, Circles, and Disks

By: Chinn, W.GContributor(s): Steenrod, N.EMaterial type: TextTextSeries: New mathematical library, 18Publication details: [New York] : Random House, ©1966Description: viii, 160 pages : illustrationsISBN: 9780883856185 ; 0883856182Subject(s): Geometry | TopologyDDC classification: 514
Contents:
Part I. Existence theorems in dimension 1 ; The first existence theorem -- Sets and functions -- Neighborhoods and continuity -- Open sets and closed sets -- The completeness of the real number system -- Compactness -- Connectedness -- Topological properties and topological equivalences -- A fixed point theorem -- Mappings of a circle into a line -- The pancake problems -- Zeros of polynomials -- Part II. Existence theorems in dimension 2 ; Mappings of a plane into itself -- The disk -- Initial attempts to formulate the main theorem -- Curves and closed curves -- Intuitive definition of winding number -- Statement of the main theorem -- When is an argument not a proof? -- The angle swept out by a curve -- Partitioning a curve into short curves -- The winding number W([small Greek phi],[small Greek gamma]) -- Properties of A([small Greek phi],[small Greek gamma]) and W([small Greek phi],[small Greek gamma]) -- Homotopies of curves -- Constancy of the winding number -- Proof of the main theorem -- The circle winds once about each interior point -- The fixed point property -- Vector fields -- The equivalence of vector fields and mappings -- The index of a vector field around a closed curve -- The mappings of a sphere into a plane -- Dividing a ham sandwich -- Vector fields tangent to a sphere -- Complex numbers -- Every polynomial has a zero -- Epilogue : a brief glance at higher dimensional cases.
Summary: 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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Includes Index

Part I. Existence theorems in dimension 1 ; The first existence theorem --
Sets and functions --
Neighborhoods and continuity --
Open sets and closed sets --
The completeness of the real number system --
Compactness --
Connectedness --
Topological properties and topological equivalences --
A fixed point theorem --
Mappings of a circle into a line --
The pancake problems --
Zeros of polynomials --
Part II. Existence theorems in dimension 2 ; Mappings of a plane into itself --
The disk --
Initial attempts to formulate the main theorem --
Curves and closed curves --
Intuitive definition of winding number --
Statement of the main theorem --
When is an argument not a proof? --
The angle swept out by a curve --
Partitioning a curve into short curves --
The winding number W([small Greek phi],[small Greek gamma]) --
Properties of A([small Greek phi],[small Greek gamma]) and W([small Greek phi],[small Greek gamma]) --
Homotopies of curves --
Constancy of the winding number --
Proof of the main theorem --
The circle winds once about each interior point --
The fixed point property --
Vector fields --
The equivalence of vector fields and mappings --
The index of a vector field around a closed curve --
The mappings of a sphere into a plane --
Dividing a ham sandwich --
Vector fields tangent to a sphere --
Complex numbers --
Every polynomial has a zero --
Epilogue : a brief glance at higher dimensional cases.

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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