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
Applications
Review

4


Oct 1st
3.2
3.3
3.4
3.5
Open sets and neighborhoods. What is "a topology"?
Continuous functions between topological spaces.
Connectedness and compactness
Separation axioms
Product spaces
Quotient spaces
5

Midterm on Wed Oct 3rd
Oct 8th


6

No class Oct 8th
Oct 15th


7


Oct 22nd


8


Oct 29th


9


Nov 5th


10


Nov 12th


11

Midterm on Friday Nov 16th  
Nov 19th


12

Nov 22-24
Thanksgiving
Nov 26th


13


Dec 3rd


14








Final

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