Fall 2010
Schedule
Syllabus
| 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 |
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| 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: |
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| 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 |
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| 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 |
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| 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. |
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| 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 |
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| 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 !!! |
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| Oct
28 Miterm |
Chapters
1-7 HISTOGRAM |
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| Nov 2 | Covering spaces |
8 | p280
#2,4 p288 #3,5 p292 #1 p298 #6 |
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| 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 |
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| Nov 9 | Covering spaces - |
9 | this problem |
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| 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) |
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| 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 |
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| Nov 18 | Special Lecture
by Dennis Sullivan - Homology Seifert - van Kampen theorem (general form) |
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| 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 |
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| 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 | |||||
