Math 551 Elementary Topology (Fall 2014)

Lectures

Office hours

Textbook

Examinations and Homework

• HW #1: Due Monday, Sep. 15: Section 1: 1, 2, 8, 10; Section 2: 1, 2, 4, 5;
• HW #2: Due Monday, Sep. 22: Section 3: 3, 4, 9; Section 5: 1; Section 6: 2, 3, 5, 6. Honors Students: Section 6: 7.
• HW #3: Due Monday, Sep. 29: Section 7: 1, 2, 3, 4. Honors students: 7: 5, 6, 7. (Honors students can turn these in later if more time needed).
• HW #4: Due Friday, Oct 17. Section 13: 1, 3, 4, 5, Section 16: 1, 3, 4, 6. Honors students: Section 13: 7, Section 16: 9, 10.
• HW #5: Due Friday, Oct 24. Section 17: 6, 8 (a) and (b), 9, 11, 12, 13. Honors students: Section 17: 8 (c), 19, 21 (Honors: first 2 due Oct 24, but number 21 due by November 26).
• HW #6: Due Friday, Oct 31. Section 18: 3, 4, 5, 6, 10, 11, 12.
• HW #7: Due Friday, November 21. Section 20: 1,3; Section 21: 2, 3, 6, 8.
• HW #8: Due Friday, December 5. Section 23: 5,9; 24: 1, 2, 3. Section 26: 3.
• HW #9: Not to turn in. Section 26: 4, 5, 6; Section 27: 2, 6.

Brief lecture outline

• Week 1: September 1-5
  
  1. Syllabus and introduction.
  2. Munkres 1-1: Elementary set theory and logic.
• Week 2: September 8-12
  
  1. 1-2: Rule of assignments, functions, domain and range.
  2. 1-2: Injective, surjective, bijective functions.
  3. 1-2: Inverse images of sets in range under functions.
  4. 1-3: Relations and equivalence relations.
  5. 1-3: Order relations, order type, dictionary order.
  6. 1-3: Least upper bound (supremum). Greatest lower bound (infimum).
• Week 3: September 15-19
  
  1. 1-4: The real numbers, well-ordering property and induction.
  2. 1-5: Finite and countable Cartesian products.
  3. 1-6: Finite sets.
  4. 1-6: Proof that cardinality of a finite set is well-defined.
  5. 1-6: Finite unions and finite cartesian products of finte sets are finite.
• Week 4: September 22 - 26
  
  1. 1-7: Infinite sets, countably infinite sets.
  2. Countable unions of countable are countable.
  3. Finite products of countable sets are countable.
  4. {0,1}^{\omega} is uncountable.
  5. The real numbers are uncountable.
  6. Power set of X can never be in bijection with X.
• Week 5: September 29 - October 3
  
  1. Exam I: Wednesday, October 1. Location Van Vleck B119 in-class.
  2. Infinite sets and the Axiom of Choice.
• Week 6: October 6 - 10
  
  1. 2-12: Definition of a topology.
  2. Discrete and trivial topologies. 29 topologies on a 3 point set.
  3. Finer and coarser topologies.
  4. 2-13: Basis for a topology.
  5. Subbasis for a topology.
  6. 2-15: The Product Topology (finite products).
  7. 2-16: The Subspace Topology.
• Week 7: October 13 - 17
  
  1. 2-17: Closed sets.
  2. Interior and closure of subsets.
  3. Limit points.
  4. Hausdorff spaces.
• Week 8: October 20 - 24
  
  1. 2-18: Continuous functions.
  2. Homeomorphisms and embeddings.
  3. Rules for continuous functions.
  4. Pasting together continuous functions.
  5. Continuous maps into product spaces.
• Week 9: October 27 - October 31
  
  1. 2-19: Arbitrary Cartesian products.
  2. Comparison of Box and Product Topologies.
  3. 2-20: The metric topology.
  4. Triangle Inequality for R^n.
  5. Comparison of metric topologies, Euclidean vs. square metric.
• Week 10: November 3 - 7
  
  1. 2-21: Continuity in metric spaces.
  2. Theorem 21.1: epsilon-delta continuity for metric spaces.
  3. Theorem 21.3: Limit point definition of continuity.
  4. Theorem 21.6: Uniform Limit Theorem.
  5. Exam II Review.
• Week 11: November 10 - 14
  
  1. 2-22: The Quotient Topology
  2. Exam II: Wednesday, November 12. Location Van Vleck B119 in-class.
  3. 3-23: Connectedness.
• Week 12: November 17 - 21
  
  1. 3-23: Closure of connected set is connected.
  2. Image of connected set under continuous map is connected.
  3. Finite Cartesian product of connected spaces is connected.
  4. 3-24: The real line is connected, and so are intervals and rays.
  5. Intermediate value theorem.
  6. Path connectedness.
  7. The topologist's sine curve.
• Week 13: November 24 - 28
  
  1. 3-26: Compactness.
  2. Closed subspace of compact space is compact.
  3. Compact subspace of Hausdorff space is closed.
  4. Image of compact set under continuous map is compact.
  5. The product of finitely many compact spaces is compact.
  6. The Tube Lemma.
  7. November 28: Thanksgiving break
• Week 14: December 1 - 5
  
  1. 3-27: Compact subspace of the real line.
  2. Cantor's nested intervals theorem.
  3. Heine-Borel theorem.
  4. Subset of R^n is compact iff closed and bounded.
  5. Extreme value theorem.
  6. 3-27: Lebesgue number lemma.
  7. Theorem 27.6: Uniform Continuity Theorem.
• Week 15: December 8 -12
  
  1. TBA
  2. Exam review.
  3. Exam III: Friday, December 12. Location Van Vleck B119.

Contact Information

Autumn Kent
Department of Mathematics
University of Wisconsin
480 Lincoln Drive
Madison, WI 53706

Office: 615 Van Vleck
Office phone: 608-263-5064
e-mail