## Math 121:  Introduction to Topology

Tentative Class Schedule

 Date Topic Homework 1/3 (Mon) Section 1.1: Definition of metric, open and closed sets A. Verify that R^n with the usual metric is a metric space (the verification of the triangle inequality is 1.1 #3 below). 1.1: 1,2,3,4,5,7,8 1/5 (Wed) Sections 1.2 and 1.3: Completeness, review of real line 1.2: 5,7  1.3: 4,7 1/7 (Fri) Section 1.4: Products, R^n 1.4: 1,3,6 1/10 Section 1.6: Continuous functions 1.6: 1,2,3,4,5 1/12 Section 1.5: Compactness, Day I A. Prove that a closed subset of a compact metric space is compact. B. Prove that a finite union of compact subsets of a metric space is compact. 1.5: 1,2,4 (in part (b), assume that all the U_\alpha's are proper subsets of X),5,8 1/14 Section 1.5: Compactness, Day II 1.6: 7,8,9,10 1/17 University Holiday (MLK Jr Day) 1/19 Sections 2.1 and 2.2:  Topological spaces, subspaces 2.1: 2,3,10,12 1/21 Section 2.3: Continuous functions A. Show that the induced topology is the smallest topology which makes f:X\to Y continuous. 2.2: 1 2.3: 3,6,12 1/24 Section 2.4: Basis for a topology 2.4: 1,4,6 1/26 Section 2.10: Finite product spaces Section 2.13: Quotient spaces 2.10: 5(a)-(d) A. Prove directly (without using Theorem 13.4) that the spaces [0,1]/~ and [0,1]^2/~ described in class with the quotient topologies are homeomorphic to S^1 and S^1\times S^1, respectively. 1/28 Section 2.5: Separation axioms, Day I 2.13: 6 (prove this without using Theorem 13.4) 2.5: 1,2,3 1/31 Section 2.5: Separation axioms, Day II 2.5: 4,8,9 2/2 Section 2.6: Compactness 2.6: 3,4,6,7,8 2/4 Section 2.7: One-point compactification 2.7: 2,3,6,7 You can turn in HW from this week on Wed 2/9. 2/7 Midterm Exam Midterm info Sample midterm problems 2/9 Section 2.8: Connectedness 2.8: 1,2,3,4,5,6 2/11 Section 2.9: Path-connectedness 2.9: 1,2,3,4,6,7 2/14 Section 2.11: Zorn's lemma 2.11: 3 2/16 Section 2.12: Infinite product spaces 2.12: 3,4,7,8,9,11 2/18 Section 3.1: Groups Section 3.2: Homotopy of paths 3.1: 1,4 2/21 University Holiday (Presidents' Day) 2/23 Section 3.2: More on homotopy of paths 3.2: 1,2,4 2/25 Section 3.3: Fundamental group 3.2: 5 3.3: 1,2,3,4 2/28 Section 3.4: Induced homomorphism 3.3: 5,6,7 3/2 Section 3.7: Homotopy of maps Section 3.5: Covering spaces, Day I 3.4: 1,2,3 2.13: 8 3/4 Section 3.5: Covering spaces, Day II 3.5: 1,3,4,5 3/7 Section 3.5: Calculation of \pi_1(S^1) Don't need to turn in 3.5: 8,9,10 3.7: 5,7 3/9 Section 3.6: Applications 3/11 Sections 3.8 and 3.9: Applications Review/summary 3/17 (Thu) Final Exam Final info Sample final problems