| Week Starts | Sections | Topics | HW | Assignment (hand in the underlined problems) on Friday. | Remarks |
|---|---|---|---|---|---|
| Aug 24th |
1.1 |
Introduction - Administrative details (all
discussed in the syllabus) Point set topology |
0 |
Fill this
form. |
|
| Sept 3rd |
2.1 2.2 |
Open and closed sets in Rn. Relative neighborhoods |
1 |
1.1 You do not need to justify your answers in this problem. This
is just to sharpen your topological neurons. Section 2.1 Open and closed sets in Rn. 2.1 (problems 1, 4, 5 ,6, 7, 8, 9.) Justify your answers. A. Determine whether the rectangle (0,2)x(0,1) is open and/or closed in R2. Justify your answer. 2.2 2.3 - (problems 4, 5 ,6, 7, 8, 9). Justify your answers. 2.4 2.9 2.12 |
Sept 3rd, Labor day. |
| Sept 10th |
2.3 2.4 |
Continuous functions. Compact sets |
2 |
Section 2.2 Relative neighborhoods For each of the following terms, down an informal definition and a formal definition. In both cases, use ONLY words and no symbols. The informal definition should be understood by a person with minimal math background (junior high level). Terms: interior point, exterior point, frontier point, limit point, open set, closed set, open set relative to a subset A, closed set relative to a subset A, (A is a subset of the plane), continuous function. In all problems below, justify all your steps. 2.19 2.22 Find the interior and the exterior of the Cantor set. Consider C=[0,1] as a subset of the real line A. Find the interior of C. Consider C=[0,1] as a subset of the Euclidean plane R^2. Find the interior of C. Prove that the intersection of finitely many open set is open. |
|
| Sept 17th |
2.5 |
Read this
blog post. Connected sets Special lecture |
3 | 2.17 Section 2.3 Continuity 2.25 B. Determine whether the function f from the interval [0,1) to the circle, defined by f(t)=(cos(2πt),sin(2πt)) is continuous, and whether the inverse of f is continuous. C. Prove that a constant function from Rn to is Rm continuous. 2.26 Section 2.4 Compact sets 2.27 2.28 D. Determine whether a circle is a compact subset of R2. |
|
| Sept 24th |
3.1 3.2 3.3 |
Open sets and neighborhoods. What is "a topology"? Continuous functions between topological spaces. Connectedness and compactness |
4 |
No homework this week, but start working for the homework for the
next. |
|
| Oct 1st |
3.4 3.5 |
Review and midterm |
5 |
3.2, 3.3 (list "all" topologies on a set of three
elements, where "all" is as we discussed in class, up to
permutation of the elements) 3.6, 3.8, 3.9, 3,10, 3,11, 3,12, 3.15, 3.18, 3.19, 3.20 |
Midterm on Wed Oct 3rd |
| Oct 8th |
Compactness, Separation axioms, Product spaces Linkages. |
6 |
3.21, 3.22, 1. Prove that every subset of the real numbers with the finite complement topology is compact. 2. Show one (or more!) compact sets that are not closed. 1. and 2. show that compact does not imply closed in general. |
No class Oct 8th |
|
| Oct 15th |
Product spaces Quotient spaces |
7 |
3.25, 3.26 3.27, 3.28, 3.29, |
||
| Oct 22nd |
Quotient spaces Surfaces Surfaces with boundary Triangulations Classification of surfaces The problems we worked on in class. Kaleydocycles: To make them, and to make more of them, variations. The long line |
8 |
3.31, 3.32, 3.33, 3.34, 3.35, 3.36 4.2, 4.8, 4.9, 4.10, 4.11, 4.12 |
||
| Oct 29th |
Classification of surfaces Topological invariants |
9 |
4.13, 4.14, 4.16. -Prove that if two surfaces homeomorphic, then one orientable if and only if the other one is. -Prove that the directed sum of two compact surfaces is orientable if and only if the two surfaces are orientable. |
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| Nov 5th |
Classification of surfaces Topological invariants |
10 |
Write down a summary (this is, definitions and main theorems) of the topics discussed in Chapter 3 and Chapter 4 of the book (Point set topology) Make sure it is clear, and contains only the essential information. Try to fit it in a small space (one or two pages). You can also add a picture that you associate with the definition or theorem. Use your own words, minimize the number of symbols. |
||
| Nov 12th |
Review Midterm |
11 |
No homework! Midterm topics: Chapters 3 and 4. |
Midterm on Friday Nov 16th | |
| Nov 19th |
Graphs and trees |
12 |
Work on the problems of the midterm.
|
Nov 22-24 Thanksgiving |
|
| Nov 26th |
The Euler characteristic Map coloring problems Graphs revisited Map coloring |
13 |
Write down a summary (this is, definitions and main theorems)
of the topics discussed in Chapter 2 of
the book (Point set topology Rn) Follow the
guidelines of the summary of Chapter 3. 5.1-5.2-5.3-5.4 Here you can adopt your very own polytope. And here is an app that illustrates Scissors Congruence. |
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| Dec 3rd |
14 |
Write down a summary of all the topics discussed in the
course, following the same guidelines of the summaries we
did before. You can reuse the previous summaries or build it from
scratch (whatever helps you learn better). 5.5-5.6-5.7- -5.11 - 5.13- 5.15 - 5.17- |
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| Final |
Final Exam: Wednesday, December 19, 2:15pm-5:00pm in our usual
classroom (Eart & Space 183) |
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