MAT 530:
Fall 2010
schedule

Topic Date Material Covered Reading Set Problem Set Due
Topological spaces generalities
and abstract stuff
(Chapters 2 and 4)
Aug 31 Topological Spaces  - Basis for a topology
Sept 2 The order Topology - Product Topology-  1.7
Sep 7 Subspace Topology-Closed sets and Limit points 1 Problem 0.
p39 #5;
p83 #3,4,7,8
HW1 Solutions
Sept 9 No class
Sep 14 Continuous Functions - Metric topology (especially, R^n, with topology induced by Euclidean metric and  order + product top.) Metric topology - (notice countable neigh. property)  18 ,  20, 21 (excluding product topology),  2 p92 #3,4,8,9,10; p100 #7,9,11,21
Problem 21 is hard. Try to find as many subsets as possible. Hint:
Sep 16 Countability axioms (see topology on R^2 with dictionary order) , The list of countablitlity and separation axioms is here.

30, 31
Sep 21 Countability axiomos examples
Separation axioms
32 , 33 3 p111 #2,6,7,8,12, 13
p126 #2,5,8, 9, 11;
p133 #2,4,6,8,9
Sept 23 Separation axioms  examples.  (hyperbolae picture -quotient space) Normal spaces  - Product spaces -  The Urysohn Lemma
19, 22,
Sep 28 Product spaces -
Finish the separation axioms diagram.
Urysohn Metrization Theorem -

34, 35
4 Topics to think about before Tuesday here
p194 #5,6.
p199 #3, 4
p205 # 1, 5, 6
Below is a list of  suggested problems not mandatory.
p199 #9
p205 #7
Homework is always due on Tuesdays
Compactness - Paracompactness Connectedness -
Produc spaces
(Chapters 3 and 5, Overview of Chapter 6)
Sept 30 Tietze Extension Theorem
Mention of Local Finiteness, Paracompactness and  Nagata-Smirnov Metrization Theorem,

23, 24
Oct 5 Connected spaces -
Connected subspaces of the real line
Components and local connectedness
24, 25 5 p118 #6, 7
p213 #3, 4
p218 #8
p223 #5,
p248 #6
Also, the problem about tangent disk topology here
There are 8 problems.
Oct 7 Compact spaces
Compact spaces of the real line
Tychonoff Theorem
26, 27, 37
Oct 12 Limit point compactness,
Sequential compactness
Countable compactness,
Compactness and Metric spaces
Some definitions are here.
28 6 p152 #10
p157 #3,8,12;
p162 #2
p170 #8,12;
p177 #3,6
Oct 14 Local compactness-
Complete metric spaces.
29, 43
Function spaces and their topologies - Ascoli theorem
(Chapter 7)
Oct 19 Compactness in Metric spaces
Pointwise and Compact Convergence
Ascoli's Theorem
45 - 46
7 p181 #1,2,6;
p186  #4, 8;
p270  #5,7,10
and this problem
There are 9 problems.
(And a cartoon about the axiom of choice)
Oct 21 Function Spaces and their topologies
Baire Spaces
47 - 48
Fundamental group -
Examples -
Van Kampen Theorem -
(Chapter 9, 10, and13)
Oct 26 Covering space construction
Homotopy of paths
The fundamental group
51 - 52
No homework !!!
Oct 28
Miterm
Chapters 1-7 HISTOGRAM
Nov 2 Covering spaces

8 p280 #2,4
p288 #3,5
p292 #1
p298 #6

Nov 4 Special Lecture by Dennis Sullivan -
Covering spaces - Classification
The fundamental group of the circle
p330 #3
p335 #4,5,7
p341 #4,5; p348 #4,5,6
Also, understand the pictures of hte coverings here
Applets for the complex exponential map  and the  map z->z^n can be found here
Nov 9 Covering spaces -
9 this problem
Nov 11 Covering transformations - Here is an interesting handout by Dror Bar Natan (note that he assumes the path and homotopy lifting theorems, and that the base spaces  of the coverings are not necesarily connected)

Nov 16 Seifert Van Kampen Theorem
Fundamental group computations
10 p483 #4
p492, #2,
#5(a) and #5(b)
Hatcher (found here) p79#9, 10,  25, 27
Nov 18 Special Lecture by Dennis Sullivan -
Homology
Seifert - van Kampen theorem (general form)
examples - surfaces - essential and inessential maps up to homotopy -
(Chapters 12)
Nov 23 Retracts, deformation retracts and homotopy type
11 p353 #1,4
p366 #4, 7, 9
Nov 25 No class - Thanksgiving.
Nov 30 Surfaces - Fundamental group  12
Dec 2 Surface - Classification - Problems to work on here
Dec 7 Surfaces - Construction 13
Dec 9 Review by Dennis Sullivan