Topology of Surfaces Book

	Preface
Ch. 1	Introduction to topology	1
1.1	An overview	1
Ch. 2	Point-set topology in R[superscript n]	7
2.1	Open and closed sets in R[superscript n]	7
2.2	Relative neighborhoods	16
2.3	Continuity	19
2.4	Compact sets	26
2.5	Connected sets	29
2.6	Applications	31
Ch. 3	Point-set topology	36
3.1	Open sets and neighborhoods	36
3.2	Continuity, connectedness, and compactness	42
3.3	Separation axioms	46
3.4	Product spaces	47
3.5	Quotient spaces	51
Ch. 4	Surfaces	55
4.1	Complexes	55
4.2	Surfaces	64
4.3	Triangulations	73
4.4	Classification of surfaces	79
4.5	Surfaces with boundary	86
Ch. 5	The euler characteristic	90
5.1	Topological invariants	90
5.2	Graphs and trees	91
5.3	The euler characteristic and the sphere	97
5.4	The euler characteristic and surfaces	104
5.5	Map-coloring problems	109
Ch. 6	Homology	116
6.1	The algebra of chains	117
6.2	Homology	128
6.3	More computations	139
6.4	Betti numbers and the euler characteristic	141
Ch. 7	Cellular functions	144
7.1	Cellular functions	144
7.2	Homology and cellular functions	152
7.3	Examples	157
7.4	Covering spaces	162
Ch. 8	Invariance of homology	170
8.1	Invariance of homology for surfaces	170
8.2	The Simplicial Approximation Theorem	174
Ch. 9	Homotopy	184
9.1	Homotopy and homology	184
9.2	The fundamental group	190
Ch. 10	Miscellany	199
10.1	Applications	199
10.2	The Jordan Curve Theorem	208
10.3	3-manifolds	212
Ch. 11	Topology and calculus	221
11.1	Vector fields and differential equations in R[superscript n]	221
11.2	Differentiable manifolds	233
11.3	Vector fields on manifolds	234
11.4	Integration on manifolds	241
	Appendix: Groups	247
	References	256
	Index	258