Tentative Schedule of Topics and Homework List

 
Date
Topic
Read
Exercises
Due
9/5
§1 Fundamental Concepts
§2 Functions
§3 Relations
p. 1-24 §1 #4-6
§2 #4,5
§3 #4
9/14
9/7
§3 Relations
§4 The Integers and the Real Numbers
§5 Cartesian Products
p. 24-38 §3 #9,15
§5 #5
9/14
9/9
§6 Finite Sets
§7 Countable and Uncountable Sets
p. 39-55 §6 #3
§7 #3-5
9/14
9/12
§9 Infinite Sets and the Axiom of Choice
§10 Well-Ordered Sets
§12 Topological Spaces
p. 57-78 §9 #5
§10 #1
List all topologies on X={a,b,c}.
9/21
9/14
§13 Basis for a Topology p. 78-82 §13 #1,3
9/21
9/16
§13 Basis for a Topology
§14 The Order Topology
p. 84-86 §13 #5,7
9/21
9/19
§15 The Product Topology on X x Y p. 86-88 §16 #4
9/28
9/21
§16 The Subspace Topology
§17 Closed Sets and Limit Points
p. 88-94 §16 #3,10
§17 #2,3
9/28
9/23
§17 Closed Sets and Limit Points p. 94-98 §17 #8,20
9/28
9/26
§18 Continuous Functions p. 102-107 §18 #4,5
10/5
9/28
§18 Continuous Functions p. 98-100,107-108 §17 #12,13
Show that if X is Hausdorff and X is homeomorphic to Y, then Y is Hausdorff.
10/5
9/30
§19 The Product Topology p. 108-115 §18 #8
10/5
10/3
§19 The Product Topology p. 115-117 §19 #4,6
10/12
10/5
§20 The Metric Topology p. 119-126 §20 #3,9
10/12
10/7
§20 The Metric Topology
§21 The Metric Topology (continued)
p. 129-133 §21 #1,2
10/12
10/10
§22 The Quotient Topology p. 136-141 §22 #2-4
10/19
10/12
§22 The Quotient Topology p. 142-144 See additional problems here.
10/19
10/14
§23 Connected Spaces
§24 Connected Subspaces of the Real Line
§25 Components and Local Connectedness
p. 147-149,153-154,159 Let X=({1/n | n is a positive integer}xI) U (Ix{0}) U {(0,1)}. Show that X is connected.
10/19
10/17
§23 Connected Spaces
§24 Connected Subspaces of the Real Line
p. 150-152,160-162 §23 #2,5,11
10/28
10/19
§24 Connected Subspaces of the Real Line p. 155-157 §24 #1,8
10/28
10/21
§26 Compact Spaces
§27 Compact Subspaces of the Real Line
p. 163-166,172-173 §26 #2,3
10/28
10/26
§26 Compact Spaces p. 166-170
10/28
§27 Compact Subspaces of the Real Line p. 174-177
11/2
§27 Compact Subspaces of the Real Line
§29 Local Compactness



11/4
§29 Local Compactness p. 182-185 §26 #5
§29 #1,3,6,8
11/11
11/7
§51 Homotopy of Paths p. 321-329 §51 #3
11/21
11/9
§51 Homotopy of Paths
§52 The Fundamental Group

Prove Theorem 51.2 parts (2) and (3) (including finding explicit path homotopies.)
11/21
11/11
§52 The Fundamental Group p. 330-334 §52 #3,4,6
11/21
11/14
§53 Covering Spaces p. 335-340

11/16
§53 Covering Spaces


11/18
§54 The Fundamental Group of the Circle p. 341-347

11/21
§54 The Fundamental Group of the Circle
§53 #3,6 (only do Hausdorff for part (a))
§54 #4,5
12/2
11/28
§55 Retractions and Fixed Points
§58 Deformation Retracts and Homotopy Type



11/30
§55 Retractions and Fixed Points
§58 Deformation Retracts and Homotopy Type



12/2
§67 Direct Sums of Abelian Groups
§68 Free Products of Groups
§69 Free Groups



12/5
§70 The Seifert-van Kampen Theorem
§71 The Fundamental Group of a Wedge of Circles

Last homework assignment.
12/12
12/7
§72 Adjoining a Two-cell


12/9
§73 The Fundamental Groups of the Torus and the Dunce Cap