Math 551 Elementary Topology (Spring 2007)

Lectures

Textbook

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Examinations and Homework

• HW #1: Due Thursday, Feb. 1. 1: 1, 2 ( don't have to turn in 2); 2: 1, 2, 4; 3: 3, 4.
• HW #2: Due Thursday, Feb. 8. 6: 2, 3, 5, 6; 7: 1, 3, 4.
• HW #3: Due Thursday, Feb. 15. 13: 1, 3, 4, 5.
• HW #4: Due Thursday, Feb. 22. 16: 1, 3, 4, 6. 17: 2,3.
• HW #5: Due Thursday, March 1. 17: 6, 8, 9, 11, 12, 13.
• HW #6: Due Thursday, March 8. 18: 3, 4, 5, 6, 10, 11, 12;
• HW #7: Due Thursday, March 29. 23: 5,9; 24: 1, 2, 3; 26: 3, 5, 6, 7, 8.
• HW #8: Due Thursday, April 12. 21: 2, 6; 26: 12; 27: 2 , 6, Extra Credit: prove part (e) without using Theorem 27.7.
• HW #9: Due Thursday, May 10. 51: 3; 52: 1, 2, 3, 4; 53: 3, 5. Everyone should do them, but only hand in for extra credit (if perfect, 10% added to final HW average).

Brief lecture outline

• Lecture 1: Tuesday, January 23
  
  1. Introduction, syllabus.
  2. Munkres 1-1: Elementary set theory and logic.
  3. 1-2: Rule of assignments, functions, domain and range.
• Lecture 2: Thursday, January 25
  
  1. 1-2: Injective, surjective, bijective functions.
  2. 1-2: Inverse images of sets in range under functions.
  3. 1-3: Equivalence relations.
• Lecture 3: Tuesday, January 30
  
  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 4: Thursday, February 1
  
  1. 1-6: Finite sets.
  2. 1-6: Proof that cardinality of a finite set is well-defined.
• Lecture 5: Tuesday, February 6
  
  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.
• Lecture 6: Thursday, February 8
  
  1. 2-12: Definition of a topology.
  2. Topologies on a 3 point set.
  3. Finer and coarser topologies.
  4. 2-13: Basis for a topology.
• Lecture 7: Tuesday, February 13
  
  1. 2-13: Basis for a topology.
  2. Subbasis for a topology.
• Lecture 8: Thursday, February 15
  
  1. 2-14: The Order Topology.
  2. 2-15: The Product Topology.
  3. 2-16: The Subspace Topology.
• Lecture 9: Tuesday, February 20
  
  1. 2-17: Closed sets.
  2. Interior and closure of subsets.
  3. Limit points.
  4. Hausdorff spaces.
• Lecture 10: Tuesday, February 27
  
  1. 2-18: Continuous functions.
  2. Homeomorphisms.
  3. Rules for continuous functions.
• Lecture 11: Thursday, March 1
  
  1. Pasting together continuous functions.
  2. Continuous maps into product spaces.
  3. 2-19: Arbitrary Cartesian products.
  4. Comparison of Box and Product Topologies.
  5. 3-23: Connectedness.
• Lecture 12: Thursday, March 8
  
  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.
• Lecture 13: Tuesday, March 13
  
  1. The topologist's sine curve.
  2. 3-26: Compactness.
  3. Exam review.
• Thursday, March 15
  
  Exam I.
• Lecture 14: Tuesday, March 20
  
  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 15: Thursday, March 22
  
  1. 3-27: Compact subspace of the real line.
  2. Cantor's nested intervals theorem.
  3. Heine-Borel theorem.
  4. Extreme value theorem.
  5. Subset of R^n is compact iff closed and bounded.
  6. Summary of exam I.
• Lecture 16: Tuesday, March 27
  
  1. 2-20: Metric spaces and the metric topology.
  2. Metrizable spaces.
  3. Various metrics on R^n, Euclidean metric, square metric, L^p metric.
  4. 2-21: Continuity in metric spaces.
• Lecture 17: Thursday, March 29
  
  1. Uniform convergence.
  2. Theorem 21.6, Uniform limit theorem.
  3. 3:27: Lebesgue number lemma.
  4. Theorem 27.6: Uniform continuity theorem.
• Lecture 18: Tuesday, April 10
  
  1. Second countable spaces.
  2. Well-ordered sets.
  3. The long line : not second countable.
  4. Manifold: second countable Hausdorff and locally Euclidean.
  5. 2-22: The Quotient topology.
  6. Example of a non-Hausdorff manifold.
  7. S^n = quotient of closed ball, identify boundary to a point.
  8. T^2 = square with sides identified appropriately.
• Lecture 19: Thursday, April 12
  
  1. Projective 2-space RP^2.
  2. Orientability.
  3. Mobius band.
  4. RP^2 is not orientable.
  5. Connect sum operation.
  6. Klein bottle = RP^2 # RP^2.
• Lecture 20: Tuesday, April 17
  
  1. Exam Review.
• Tuesday, April 19
  
  Exam II.
• Lecture 21: Tuesday, April 24
  
  1. Classification Theorem for compact surfaces.
  2. Examples of various 2n-gons with boundary identifications.
  3. Triangulations of surfaces.
  4. Basic idea of proof of classification theorem.
• Lecture 22: Tuesday, May 1
  
  1. Discussion of Exam II.
  2. K = P^2 # P^2.
  3. T^2 # P^2 = P^2 # P^2 # P^2.
  4. Triangulations of surfaces.
  5. Euler characteristic of surfaces X = V - E + F.
  6. X(S^2) = 2, X(RP^2) = 1, X(T^2) = 0.
  7. X(S_1 # S_2) = X(S_1) + X(S_2) - 2.
  8. X(g#T^2) = 2 - 2g, X(g# P^2) = 2 - g.
• Lecture 23: Thursday, May 3
  
  1. 9-51: Homotopy of paths.
  2. 9-52: The fundamental group.
• Lecture 24: Tuesday, May 8
  
  1. 9-53: Covering spaces.
  2. Simply connected covering space = universal cover.
  3. \pi_1(B, b_0) = fiber of universal cover.
• Lecture 25: Thursday, May 10
  
  1. Applications of fundamental group and covering spaces.
  2. R^2 and R^n are not homeomorphic for n > 2.
  3. Final exam review.
• Sunday, May 13
  
  Final Exam: 12:25PM - 2:25PM VAN VLECK B215.