Play-by-Play Math 501

References:

Royden (R )

Goldberg (G)

Bartle – Sherbert (B-S).

• indicates particularly important question.

Cl 1: Wed 9.03.08 Introduction to issues of Real Analysis via Chaotic and Regular billiards.

C2: Friday 9.05.08 - Cardinality (Royden Ch 1.2, 1.6; G Ch 1.5; B-S Ch 1.3, Adx B).

HW: S-B, Ch. 1.3 #4, 8, 9, Prove that a countable union of countable sets is countable. G, Ch 1.5 , #7, 10, 11

- Equivalence Relations (G Ch. 1.5)

o HW: Show that ‘same size’ (equivalent sets) is an equivalence relation. Give an example of sameness from another area of math and show it is an equivalence relation.

- Metric Space (Ch 7, R; various undergraduate analysis texts). Open, Closed sets. (also Ch 2.5, R for real numbers).

o HW: R, Ch. 2.5 # 24, 25, 26, 27*, 28, 29, 30, 31, 32, 33, 34 (hard)

o R. Cj 7.2 #5, 6,

o Prove Proposition 14, Ch. 2.5: The complement of an open set is closed and the complement of a closed set is open.

Continuous Functions: Sect 7.4, Sect 2.6

Heine-Borel Theorem (R, Section 2.5; B-S p. 321). Sets in R¹ (Rⁿ) closed and bounded iff compact.

Continuous function on compact sets are bounded, attain max/min.

HW #2. Due Monday Sept 29.