|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
Continuous functions between topological spaces.
Connectedness and compactness
||Midterm on Wed Oct 3rd
||No class Oct 8th
||Midterm on Friday Nov 16th|
||Final Exam: Wednesday, December 19, 2:15pm-5:00pm