Tentative Schedule of Topics and Homework List

 
Date
Topic
Read
Exercises
Due
9/4
§1 Fundamental Concepts
§2 Functions
§3 Relations
p. 1-28 §1 #4-6
§2 #4,5
§3 #9,15
9/6
9/6
§4 The Integers and the Real Numbers
§5 Cartesian Products
§6 Finite Sets
p. 30-43 §5 #5
§6 #1,3
9/13
9/11
§7 Countable and Uncountable Sets
§9 Infinite Sets and the Axiom of Choice
§10 Well-Ordered Sets
p. 44-50 §7 #2,3,5
9/13
9/13
§12 Topological Spaces
§13 Basis for a Topology
Skim p. 57-91
p. 75-83
List all the topologies on X={a,b,c}.
§13 #3,5,7
9/20
9/18
§14 The Order Topology
§15 The Product Topology on X x Y
§16 The Subspace Topology
p. 84-91 §16 #3,4,10
9/27
9/20
§17 Closed Sets and Limit Points p. 92-100 §17 #3,8,13,20
9/27
9/25
§18 Continuous Functions p. 102-111 §18 #4,5,8
Show that if X is Hausdorff and X is homeomorphic to Y, then Y is Hausdorff.
10/4
9/27
§19 The Product Topology p. 112-117 §19 #5
10/4
10/2
§20 The Metric Topology
§21 The Metric Topology (continued)
p. 119-133 §20 #3,8a
10/11
10/4
§22 The Quotient Topology p. 136-144 §22 #2-4
10/11
10/9
Cell Complexes


10/11
Manifolds
Midterm handed out
10/18
10/18
§23 Connected Spaces p. 147-152 §23 #2,3,5,11
Let X=({1/n | n is a positive integer}xI) U (Ix{0}) U {(0,1)}. Show that X is connected.
10/25
10/23
§24 Connected Subspaces of the Real Line
§25 Components and Local Connectedness
p. 153-162 §24 #1,8
§25 #8
11/1
10/25
§26 Compact Spaces p. 163-170 §26 #2-5
11/1
10/30
§27 Compact Subspaces of the Real Line p. 172-177 §27 #6
11/8
11/1
§28 Limit Point Compactness
§29 Local Compactness
p. 178-185 §28 #3a
§29 #1,3,6
11/8
11/6
§30 The Countability Axioms
§31 The Separation Axioms
§32 Normal Spaces
p. 189-204 §30 #4,12
§31 #1
§36 #5
11/15
11/8
§51 Homotopy of Paths p. 321-329 §51 #3
Prove Theorem 51.2 (including finding explicit path homotopies.)
11/15
11/13
§52 The Fundamental Group p. 330-334 §52 #3,4,6
11/29
11/15
§55 Retractions and Fixed Points p. 348-353 §55 #4a
11/29
11/20
§53 Covering Spaces p. 335-340 §53 #3,6b
11/29
11/27
§54 The Fundamental Group of the Circle p. 341-347 §54 #4,5
12/6
11/29
§58 Deformation Retracts and Homotopy Type p. 359-365 §58 #2,5,8
12/6
12/4
Overview of Seifert-Van Kampen Theorem


12/6
Overview of Covering Spaces and Deck Transformations