1.1:
Open and closed sets
1.2:
Continuous functions
1.3:
Some topological properties
1.4: A
brief introduction to dimension (optional)
CHAPTER 2: SURFACES
2.1:
Definition of a surface
2.2:
Connected sum construction
2.3:
Plane models of surfaces
2.4:
Orientability
2.5:
Plane models of nonorientable surfaces
2.6:
Classification of surfaces
2.7:
Proof of the classification theorem for surfaces (optional)
CHAPTER 3: THE EULER
CHARACTERISTIC
3.1:
Cell complexes and the Euler characteristic
3.2:
Triangulations
3.3 Genus
3.4:
Regular complexes
3.5: b-valent complexes
CHAPTER 4: MAPS AND
GRAPHS
4.1:
Maps and map coloring
4.2:
The five-color theorem for S2
4.3:
Introduction to graphs
4.4:
Graphs in surfaces
4.5:
Embedding the complete graphs, and graph coloring
CHAPTER 5: VECTOR
FIELDS ON SURFACES
5.1:
Vector fields in the plane
5.2: Index of a critical point
5.3: Limit sets in the plane
5.4: A local description of a
critical point
5.5: Vector fields on surfaces
CHAPTER 6: THE
FUNDAMENTAL GROUP
6.1:
Path homotopy and the fundamental group
6.2:
The fundamental group of the circle.
6.3:
Deformation retracts
6.4:
Further calculations
6.5:
Presentations of groups
6.6:
The Seifert-van Kampen Theorem and the fundamental groups of surfaces
6.7: Proof of the Seifert-van
Kampen Theorem
CHAPTER 7:
INTRODUCTION TO KNOTS
7.1:
Knots—what they are and how to draw them
7.2:
Prime knots
7.3:
Alternating knots
7.4:
Reidemeister moves
7.5:
Some simple knot invariants
7.6:
Surfaces with boundary
7.7.
Knots and surfaces
7.8: Knot polynomials