## Math 121:  Introduction to Topology

Tentative Class Schedule

 Date Topic Homework 4/1 (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 4/3 (Wed) Sections 1.2 and 1.3: Completeness, review of real line 1.2: 5,7  1.3: 4,7 4/5 (Fri) Section 1.4: Products, R^n 1.4: 1,3,6 4/8 Section 1.6: Continuous functions 1.6: 1,2,3,4,5 4/10 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 4/12 Section 1.5: Compactness, Day II 1.6: 7,8,9,10 4/15 Sections 2.1 and 2.2:  Topological spaces, subspaces 2.1: 2,3,10,12 4/17 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 4/19 Section 2.4: Basis for a topology 2.4: 1,4,6 4/22 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. 4/24 Section 2.5: Separation axioms, Day I 2.13: 6 (prove this without using Theorem 13.4) 2.5: 1,2,3 4/26 Section 2.5: Separation axioms, Day II 2.5: 4,8,9 4/29 Section 2.6: Compactness 2.6: 3,4,6,7,8 5/1 Section 2.7: One-point compactification 2.7: 2,3,6,7 5/3 Midterm Exam Midterm info Sample midterm problems 5/6 Section 2.8: Connectedness 2.8: 1,2,3,4,5,6 5/8 Section 2.9: Path-connectedness 2.9: 1,2,3,4,6,7 5/10 Section 2.11: Zorn's lemma 2.11: 3 5/13 Section 2.12: Infinite product spaces 2.12: 3,4,7,8,9,11 5/15 Section 3.1: Groups Section 3.2: Homotopy of paths 3.1: 1,4 5/17 Section 3.2: More on homotopy of paths 3.2: 1,2,4 5/20 Section 3.3: Fundamental group 3.2: 5 3.3: 1,2,3,4 5/22 Section 3.4: Induced homomorphism 3.3: 5,6,7 5/24 Section 3.7: Homotopy of maps Section 3.5: Covering spaces, Day I 3.4: 1,2,3 2.13: 8 5/27 University Holiday (Memorial Day) 5/29 Section 3.5: Covering spaces, Day II 3.5: 1,3,4,5 5/31 Section 3.5: Calculation of \pi_1(S^1) 3.5: 8,9,10 3.7: 5,7 6/3 Section 3.6: Applications 6/5 Sections 3.8 and 3.9: Applications 6/7 Review/summary 6/10 (Mon) Final Exam 3-6pm Location: Geology 4645 (the usual classroom) Final info Sample final problems