A Geometric Introduction to Topology

Description

PART 0 PRELIMINARIES Chapter 0 Notations and Prerequisites Numbers Sets and maps Equivalence relations Chapter 1 Spaces and Continuous Maps Introduction Continuity Homeomorphism Neighborhoods, open and closed sets Compactness Chapter 2 Abelian Groups Introduction Definitions Direct sums Exact sequences Free abelian groups PART 1 INTRODUCTION TO HOMOTOPY THEORY Chapter 3 Connected and Disconnected Spaces Introduction Connectedness Path-connectedness Local path-connectedness Chapter 4 More about Connection Introduction The group H0(X) The set π0(X) The group H0(X) Chapter 5 Definition of Homotopy Introduction Definition of homotopy Homotopy equivalence Homotopy sets; the groups H1(X) Chapter 6 A Study of a Circle Introduction Lifting maps from S1 up to R The degree of a map Applications Chapter 7 Lifting and Extension Problems Introduction The lifting problem The extension problem Chapter 8 Calculations Introduction The Mayer-Vietoris theorem First calculations Graphs Products PART 2 THE DUALITY THEOREM Chapter 9 Eilenberg's Separation Criterion Introduction Complementary components Separation of points by compact plane sets Chapter 10 The Duality Map Introduction Construction of the duality map Proof of injectivity Chapter 11 Proof of the Duality Theorem Introduction An extension theorem Naturality Proof in some special cases End of the proof Chapter 12 Remarks on the Proof Introduction The extended plane Reformulation of preceding chapters The Hopf map PART 3 FURTHER RESULTS IN THE TOPOLOGY OF PLANE SETS Chapter 13 The Jordan Curve Theorem Introduction Theta curves First alternative proof (after Dieudonne) Point sets in Rn and Sn Second alternative proof (after Doyle) Invariance of (plane) domains Chapter 14 Further Duality Properties Introduction The group H1(X) Properties of Hl(X) Duality Plane domains Chapter 15 Geometric Integration Theory Introduction Line integrals in R2 Green's theorem Reformulation in terms of homology The three-dimensional case The complex case Index of Terms Index of Notation