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# A Combinatorial Introduction to Topology by Henle

*A Combinatorial Introduction to Topology* by Henle

**Contents**

Chapter One Basic Concepts

1 The combinatorial Method

2 Continuous Transformations in the Plane

3 Compactness and Connectedness

4 Abstract Point Set Topology

Chapter Two Vector Fields

5 A link Between Analysis and Topology

6 Sperner's Lemma and the Brouwer Fixed Point Theorem

7 Phase Portraits and the Index Lemma

8 Winding Numbers

9 Isolated Critical Points

10 The Poincare Index Theorem

11 Closed Integral Paths

12 Further Results and Applications

Chapter Three Plane Homology and the Jordan Curve Theorem

13 Polygonal Chains

14 The Algebra of Chains on a Grating

15 The Boundary Operator

16 The Fundamental Lemma

17 Alexander's Lemma

18 Proof of the Jordan Curve Theorem

Chapter Four Surfaces

19 Examples of Surfaces

20 The Combinatorial Definition of a Surface

21 The Classification Theorem

22 Surfaces with Boundary

Chapter Five Homology of Complexes

23 Complexes

24 Homology Groups of a Complex

25 Invariance

26 Betti Numbers and the Euler Characteristic

27 Map Coloring and Regular Complexes

28 Gradient Vector Fields

29 Integral Homology

30 Torsion and Orientability

31 The Poincare Index Theorem Again

Chapter Six Continuous Transformations

32 Covering Spaces

33 Simplicial Transformations

34 Invariance Again

35 Matrixes

36 The Lefschetz Fixed Point Theorem

37 Homotopy

38 Other Homologies

Chapter Seven Topics in Point Set Topology

39 Cryptomorphic Versions of Topology

40 A Bouquet of Topological Properties

41 Compactness Again

42 Compact Metric Spaces