Math 551 Elementary Topology (Fall 2010)

Lectures

Textbook

Graders

Mailing List

Examinations and Homework

• HW #1: Due Thursday, Sep. 16. 1: 1, 10; 2: 1, 2, 4; 3: 3, 4. Honors students: 1: 2, 2: 5.
• HW #2: Due Friday, Sep. 24. 6: 2, 3, 5, 6; 7: 1, 3, 4. Honors students: 6:7; 7: 5.
• HW #3: Due Friday, Oct 1. 13: 1, 3, 4, 5. Honors students: 13: 7.
• HW #4: Due Friday, Oct 8. 16: 1, 3, 4, 6. Honors students: 9.
• HW #5: Due Friday, Oct 15. 17: 6, 8 (a) and (b), 9, 11, 12, 13. Honors students: 21.
• HW #6: Due Friday, Oct 22. 18: 3, 4, 5, 6, 10, 11, 12; Honors students: 17: 8 (c).
• HW #7: Due Friday, Oct 29. 23: 5,9; 24: 1, 2, 3.
• HW #8: Due Friday, Nov 5. 26: 3, 5, 6, 7, 8.
• HW #9: Due Friday, Dec 3. 21: 2, 6; 26: 4, 12; 27: 2, 4, 6. Honors students: directly prove 27:6(e) without using Theorem 27.7.

Brief lecture outline

• Lecture 1: Thursday, September 2
  
  1. Syllabus and introduction.
• Lecture 2: Tuesday, September 7
  
  1. Munkres 1-1: Elementary set theory and logic.
  2. 1-2: Rule of assignments, functions, domain and range.
• Lecture 3: Thursday, September 9
  
  1. 1-2: Injective, surjective, bijective functions.
  2. 1-2: Inverse images of sets in range under functions.
  3. 1-3: Equivalence relations.
• Lecture 4: Tuesday, September 14
  
  1. 1-3: Order relations.
  2. 1-3: Least upper bound (supremum). Greatest lower bound (infimum).
  3. 1-4: Induction.
  4. 1-5: Finite and countable Cartesian products.
• Lecture 5: Thursday, September 16
  
  1. 1-6: Finite sets.
  2. Proof that cardinality of a finite set is well-defined.
• Lecture 6: Tuesday, September 21
  
  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.
• Lecture 7: Thursday, September 23
  
  1. Power set of X can never be in bijection with X.
  2. 2-12: Definition of a topology.
  3. Finer and coarser topologies.
  4. 29 topologies on a 3 point set.
• Lecture 8: Tuesday, September 28
  
  1. 2-13: Basis for a topology.
  2. Subbasis for a topology.
• Lecture 9: Thursday, September 30
  
  1. 2-14: The Order Topology.
  2. 2-15: The Product Topology.
  3. 2-16: The Subspace Topology.
• Lecture 10: Tuesday, October 5
  
  1. 2-17: Closed sets.
  2. Interior and closure of subsets.
  3. Exam review.
• Thursday, October 7
  
  1. Exam I. 13:00-14:15, Location: Van Vleck B119.
• Lecture 11: Tuesday, October 12
  
  1. Limit points.
  2. Hausdorff spaces.
  3. 2-18: Continuous functions.
• Lecture 12: Thursday, October 14
  
  1. 2-18: Continuous functions.
  2. Homeomorphisms.
  3. Rules for continuous functions.
  4. Pasting together continuous functions.
  5. Continuous maps into product spaces.
• Lecture 13: Tuesday, October 19
  
  1. 2-19: Arbitrary Cartesian products.
  2. Comparison of Box and Product Topologies.
  3. 3-23: Connectedness.
• Lecture 14: Thursday, October 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.
• Lecture 15: Tuesday, October 26
  
  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.
• Lecture 16: Thursday, October 28
  
  1. 3-26: Compactness.
  2. Closed subspace of compact space is compact.
  3. Compact subspace of Hausdorff space is closed.
• Lecture 17: Tuesday, November 2
  
  1. Image of compact set under continuous map is compact.
  2. The product of finitely many compact spaces is compact.
  3. The Tube Lemma.
• Lecture 18: Thursday, November 4
  
  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.
• Lecture 19: Tuesday, November 9
  
  1. Exam review.
• Thursday, November 11
  
  1. Exam II. 13:00-14:15, Location: Van Vleck B119.
• Lecture 20: Tuesday, November 16
  
  1. 2-20: The metric topology.
  2. Triangle Inequality for R^n.
  3. Comparison of metric topologies, Euclidean vs. square metric.
• Lecture 21: Thursday, November 18
  
  1. Introduction to groups and fundamental group.
• Lecture 22: Tuesday, November 23
  
  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.
• Thursday, November 25
  
  Thanksgiving vacation.
• Lecture 23: Tuesday, November 30
  
  1. 3.27: Lebesgue number lemma.
  2. Theorem 27.6: Uniform Continuity Theorem.
  3. Second Countable spaces.
  4. The long line.
  5. Manifold: second countable, Hausdorff, locally Euclidean.
  6. S^1 is a manifold (stereographic projection).
• Lecture 24: Thursday, December 2
  
  1. S^n is a manifold (stereographic projection).
  2. 22: The quotient topology.
  3. Torus as quotient of square.
  4. RP^n: real projective space.
  5. A non-Hausdorff "manifold".
  6. Orientations.
  7. RP^2 and Mobius band not orientable.
• Lecture 25: Tuesday, December 7
  
  1. Connected sums of manifolds.
  2. Klein bottle = RP^2 # RP^2.
  3. Classification Theorem for compact surfaces.
  4. Examples of various 2n-gons with boundary identifications.
  5. Triangulations of surfaces.
• Lecture 26: Thursday, December 9
  
  1. Classification theorem for surfaces.
  2. T^2 # P^2 = P^2 # P^2 # P^2.
  3. Euler characteristic of surfaces X = V - E + F.
  4. X(S^2) = 2, X(RP^2) = 1, X(T^2) = 0.
  5. X(S_1 # S_2) = X(S_1) + X(S_2) - 2.
  6. X(g#T^2) = 2 - 2g, X(g# P^2) = 2 - g.
• Lecture 27: Tuesday, December 14
  
  1. Exam III. 13:00-14:15, Location: Van Vleck B119.

Contact Information

Dr. Jeff Viaclovsky
Department of Mathematics
University of Wisconsin
480 Lincoln Drive
Madison, WI 53706

Office: 803 Van Vleck
Office phone: 608-263-1161
e-mail: jeffv@math.wisc.edu