Topology lecture notes

Ward T.

Table of contents :
Topological Spaces……Page 5
The subspace topology……Page 7
The product topology……Page 8
Three important examples of quotient topologies……Page 11
Examples……Page 14
Hausdorff Spaces……Page 15
Examples……Page 16
Path connectedness……Page 20
Homotopy equivalence……Page 22
The Fundamental Group……Page 28
Based Maps……Page 29
Moving the base point……Page 30
Covering spaces……Page 33
Lifting maps……Page 35
The action on the fibre……Page 37
Classification of surfaces……Page 41
Orientation……Page 44
Polygonal representation……Page 46
Transformation to standard form……Page 47
Juxtaposition of symbols……Page 51
Euler characteristic……Page 53
Invariance of the characteristic……Page 54
Simplicial complexes and Homology groups……Page 56
Chains, cycles and boundaries……Page 57
The equation 2=0……Page 60
More homology calculations……Page 61
Geometrical interpretation of homology……Page 62
Euler characteristic……Page 67
Simplicial approximation and an application……Page 68
Chain complexes and mappings……Page 71
Relative homology……Page 72
The exact homology sequence of a pair……Page 74
Finitely generated abelian groups……Page 79
The Fundamental Theorem……Page 80
Exact sequences……Page 81
Review problems……Page 83