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Topology
Sam Macdonald
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You can't use 'macro parameter character #' in math mode
You can't use 'macro parameter character #' in math mode
Front Matter
Colophon
How to Use This Book (*)
1
Preliminaries
1.1
Overview of the Course
1.2
Useful Tools on Sets and Functions
1.2.1
Properties of Sets
1.2.2
Properties of functions
1.2.3
Constructions of new sets from old ones, and the associated functions
Subsets
Product sets
Quotient sets and equivalence relations
1.2.4
Cardinality
1.2.5
Upper and lower bound properties of
N
and
R
2
Topological Spaces and Continuous Functions
2.1
Topology and Continuity
2.1.1
Topology
2.1.2
Continuity
2.1.3
Bases
2.2
Constructing New Spaces and Continuous Functions from Old Ones
2.2.1
Subspaces
2.2.2
Product spaces
2.2.3
Quotient = identification spaces
2.3
Closed Sets, Boundaries, and Continuity
2.3.1
Closed sets
2.3.2
Closures, interiors, boundaries, and limit points
3
Homeomorphism Invariants
3.1
Motivation and Hausdorff
3.2
Metrizability
3.3
Connectedness
3.3.1
Connected
3.3.2
Path connected
3.3.3
Components
3.4
Compactness
3.5
Separation and Countable Basis Properties
3.5.1
Countable Basis
3.5.2
Separation properties
3.5.3
Interactions among homeomorphism invariants
4
Homotopy
4.1
Overview of Algebraic Topology
4.2
Retracts
4.3
Homotopy and Homotopy Equivalence
5
Fundamental Groups
5.1
Definition of
π
1
5.2
Group homomorphisms
5.2.1
Change of basepoint and PC spaces
5.2.2
Homomorphisms induced by continuous maps
5.2.3
5.2.4
Interactions with constructions
5.3
π
1
(
S
1
)
5.4
Presenting and Decomposing Groups
5.4.1
Review and presentations
5.4.2
Building new groups from old -- or decomposing groups
Abelianization
Direct products
Free Products
Free products with amalgamation
5.5
SVK
5.5.1
SVK Theorem statement and first examples
5.5.2
Classification of surfaces
5.5.3
Proof and corollaries of the Seifert-Van Kampen Theorem
5.6
Presenting Spaces and the 2-way Street Theorem
5.6.1
CW complexes
5.6.2
Fundamental groups of CW complexes
6
Covering Spaces
6.1
Definitions and Lifting
6.1.1
Definition
6.1.2
Lifting Theorems
6.1.3
Application to group theory
6.1.4
The number of sheets
6.1.5
Interactions with functions and constructions:
6.1.6
Interactions with homeomorphism invariants:
6.2
Building Covering Spaces Using Group Actions
6.2.1
Covering space group actions
6.2.2
Building SC covering spaces from group presentations
6.2.3
Existence of covering spaces
6.2.4
Building covering spaces for subgroups from group presentations
6.2.5
Applications to group theory
6.3
The Universal Covering and Galois Correspondence
7
Homology
7.1
Simplicial Homology
7.1.1
Overview of Homotopy and Homology
7.1.2
\delta -Complexes
7.1.3
Simplicial Homology
7.2
Singular Homology
7.2.1
Definitions and Induced Homomorphisms
7.2.2
Decomposing Spaces and Homology Groups
Backmatter
🔗
Chapter
4
Homotopy
4.1
Overview of Algebraic Topology
4.2
Retracts
4.3
Homotopy and Homotopy Equivalence
