Welcome to Math 4320 Reall Analysis II. Take a look at the course syllabus, and if you're ready you can start on Homework #1.. Homework will (usually) be posted here on Monday or Tuesday, and due in class the following Monday (unless stated otherwise). We will usually go over the homework problems on the day they are due, but I will not collect or grade your homework solutions. Sometimes there will be a short quiz after we discuss the homework which will be graded.
Section 10.1 #2,7
Section 10.2 #1,2
Section 10.3 #1,2,7,11
Section 11.1 #4,5,6,9
Section 11.2 #1,6,7,9,10
Section 12.1 #3,4,5
Section 12.2 # 1,2,8,9,10
Also, according to the syllabus, Exam #1 is Wednesday, February 21. The exam will cover the material we covered in chapters 10,11,12, including the "bonus material" on the topological definition of compactness. Monday, 2/19 will be a review day.
1) What is the boundary of the set Q = rational numbers (subset of R^1)?
2) Prove that x is on the boundary of A if and only if for every epsilon, the open ball B_epsilon(x) intersects A and A^c (A compliment).
3) Use the topological definition of compactness (via open covers) to prove that any closed subset of a compact set is compact. (Don't use the closed and bounded charactarization of compact subsets in Euclidian space.)