MAT 364 Topology and Geometry

Fall 2018 - Schedule

 

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.
  •  Here is a general description.
  • Here is a discussion of four bar linkages.
  • Here is an article about linkages and surfaces.
  • Here is an extraordinaire articles that discusses 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.

Nov 5th

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. 

Nov 12th


Review
Midterm
11
No homework!
Midterm topics: Chapters 3 and 4.
Midterm on Friday Nov 16th  
Nov 19th


The Euler characteristic
Map coloring problems
Graphs revisited
Map coloring
12

Nov 22-24
Thanksgiving
Nov 26th


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.



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).
Final

Final Exam: Wednesday, December 19, 2:15pm-5:00pm