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 hun...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autores principales: Chinn, William G. (Autor), Steenrod, Norman Earl, 1910-1971 (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Cambridge : Cambridge University Press, 2012.
Colección:Anneli Lax new mathematical library.
Temas:
Topology.
Geometry.
Topologie.
Géométrie.
geometry.
Geometry
Topology
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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.

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