|Week Starts||Sections||Topics||HW||Assignment (hand in the underlined problems) on Friday.||Remarks|
||Introduction - Administrative details (all
discussed in the syllabus)
Point set topology
|Open and closed sets in Rn.
||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.
2.3 - (problems 4, 5 ,6, 7, 8, 9). Justify your answers.
|Sept 3rd, Labor day.
||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.
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.
Section 2.3 Continuity
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.
Section 2.4 Compact sets
D. Determine whether a circle is a compact subset of R2.
|Open sets and neighborhoods. What is "a topology"?
Continuous functions between topological spaces.
Connectedness and compactness
||No homework this week, but start working for the homework for the
Review and midterm
||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
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
||3.25, 3.26 3.27, 3.28, 3.29,
Surfaces with boundary
Classification of surfaces
The problems we worked on in class.
Kaleydocycles: To make them, and to make more of them, variations.
The long line
||3.31, 3.32, 3.33, 3.34, 3.35, 3.36
4.2, 4.8, 4.9, 4.10, 4.11, 4.12
||Classification of surfaces
||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.
||Graphs and trees||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.
Midterm topics: Chapters 3 and 4.
|Midterm on Friday Nov 16th|
The Euler characteristic
Map coloring problems
||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.
|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).|
||Final Exam: Wednesday, December 19, 2:15pm-5:00pm