1. MA4266 Topology Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@nus.edu.sgmatwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1 Lecture 4.

2. Continuity Thm 3.13 Let is continuous when be metric spaces. A function 1. definition on page 75, any of the following equivalent conditions hold: 2. sequence ---, 3. open sets ---, 4. closed sets ---.

3. Equivalence Definition Let is be metric spaces. A function 1. an isometry if ---, 2. a homeomorphism if ---. Examples Definition Uniform continuity and uniform equivalence.

4. New Spaces from Old Definition Subspaces --- Hilbert Space Definition Product metric space ---.

5. Complete Metric Spaces Definition Cauchy Sequences --- Thm 3.16 A subspace (A,d) of a complete Definition Complete Metric Space ---, metric space (X,d) is complete iff A is closed.

6. Baire Category Definition Nowhere dense --- Definition 1 st category and 2 nd category space Thm 3.17 Baire Category Theorem. Every complete metric space is of second category. Thm 3.18 Baire Category Theorem. Every complete metric space is of second category. Definition Contractive functions ---

7. Assignment 4 Read pages 82-97 in Chapter 3 Written Homework Problems Due Tues 1 February Solve all problems in Exercises 1. Do problem 4 on page 77 2. Do problem 6 on page 82 3. Do problem 5 on page 86 4. Do problem 7 on page 95 TEST 1. Tuesday 15 February 2011