Math 551 Elementary Topology (Fall 2013)
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Lectures
Lectures are by
Jeff Viaclovsky on Mondays, Wednesday and Fridays
at 12:05-12:55 PM in Van Vleck B119. I will have office hours on Wednesdays
in Van Vleck 803 from 14:30 - 15:30 PM.
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Textbook
Topology (2nd Edition) by James Munkres.
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Graders
The grader is Carolyn Abbott. Email: abbott "at" math.wisc.edu.
Office: 820 Van Vleck. Phone: 262-7398.
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Mailing List
The class mailing list is math551-1-f13 "at" lists.wisc.edu.
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Examinations and Homework
There will be several homework assignments, and 3 in-class
exams. The final grade will be roughly 25% for each in-class
exam, and 25% for homework.
The in-class exams are scheduled for October 4, November 13,
and December 13.
Homework is due at the beginning of class on the
due date. No late homework will be accepted, unless you have
a note from the dean.
- HW #1: Due Monday, Sep. 16: Section 1: 1, 2, 8, 10; Section 2: 1, 2, 4, 5;
- HW #2: Due Monday, Sep. 23: Section 3: 3, 4, 9; Section 5: 1; Section 6: 2, 3, 5, 6.
Honors Students: Section 6: 7.
- HW #3: Due Monday, Sep. 30: 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 Wednesday, Oct 9. Section 13: 1, 3, 4, 5. Honors students: Section 13: 7.
- HW #5: Due Wednesday, Oct 16. Section 16: 1, 3, 4, 6. Honors students:
Section 16: 9, 10.
- HW #6: Due Wednesday, Oct 23. Section 17: 6, 8 (a) and (b), 9, 11, 12, 13.
Honors students: Section 17: 8 (c), 19, 21
(Honors: first 2 due Oct 23, but number 21 due by November 27).
- HW #7: Due Wednesday, Oct 30. Section 18: 3, 4, 5, 6, 10, 11, 12.
- HW #8: Due Friday, November 8. Section 20: 1,3; Section 21: 2, 6.
- HW #9: Due Monday, November 25. Section 23: 5,9; 24: 1, 2, 3.
- HW #10: Due Friday, December 6. Section 26: 3, 4, 5, 6, 7, 8; Section 27: 2, 6.
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Brief lecture outline
- Week 1: September 2-6
- Syllabus and introduction.
- Munkres 1-1: Elementary set theory and logic.
- Week 2: September 9 - 14
- 1-2: Rule of assignments, functions, domain and range.
- 1-2: Injective, surjective, bijective functions.
- 1-2: Inverse images of sets in range under functions.
- 1-3: Relations and equivalence relations.
- 1-3: Order relations, order type, dictionary order.
- 1-3: Least upper bound (supremum). Greatest lower bound (infimum).
- Week 3: September 16 - 20
- 1-4: The real numbers, well-ordering property and induction.
- 1-5: Finite and countable Cartesian products.
- 1-6: Finite sets.
- 1-6: Proof that cardinality of a finite set is well-defined.
- 1-6: Finite unions and finite cartesian products of finte sets are finite.
- Week 4: September 23 - 27
- 1-7: Infinite sets, countably infinite sets.
- Countable unions of countable are countable.
- Finite products of countable sets are countable.
- {0,1}^{\omega} is uncountable.
- The real numbers are uncountable.
- Power set of X can never be in bijection with X.
- 2-12: Definition of a topology.
- Discrete and trivial topologies. 29 topologies on a 3 point set.
- Week 5: September 30 - October 4
- Finer and coarser topologies.
- 2-13: Basis for a topology.
- Exam I: Friday, October 4. Location Van Vleck B119 in-class.
- Week 6: October 7 - 11
- Subbasis for a topology.
- 2-14: The Order Topology.
- 2-15: The Product Topology (finite products).
- 2-16: The Subspace Topology.
- Week 7: October 14 - 18
- 2-17: Closed sets.
- Interior and closure of subsets.
- Limit points.
- Hausdorff spaces.
- Week 8: October 21 - 25
- 2-18: Continuous functions.
- Homeomorphisms and embeddings.
- Rules for continuous functions.
- Pasting together continuous functions.
- Continuous maps into product spaces.
- Week 9: October 28 - November 1
- 2-19: Arbitrary Cartesian products.
- Comparison of Box and Product Topologies.
- 2-20: The metric topology.
- Triangle Inequality for R^n.
- Comparison of metric topologies, Euclidean vs. square metric.
- Week 10: November 4 - 8
- 2-21: Continuity in metric spaces.
- Theorem 21.1: epsilon-delta continuity for metric spaces.
- Theorem 21.3: Limit point definition of continuity.
- Theorem 21.6: Uniform Limit Theorem.
- Exam II Review.
- Week 11: November 11 - 15
- 2-22: The Quotient Topology
- Exam II: Wednesday, November 13. Location Van Vleck B119 in-class.
- 3-23: Connectedness.
- Week 12: November 18 - 22
- 3-23: Closure of connected set is connected.
- Image of connected set under continuous map is connected.
- Finite Cartesian product of connected spaces is connected.
- 3-24: The real line is connected, and so are intervals and rays.
- Intermediate value theorem.
- Path connectedness.
- The topologist's sine curve.
- Week 13: November 25 - 29
- 3-26: Compactness.
- Closed subspace of compact space is compact.
- Compact subspace of Hausdorff space is closed.
- Image of compact set under continuous map is compact.
- The product of finitely many compact spaces is compact.
- The Tube Lemma.
- November 29: Thanksgiving vacation
- Week 14: December 2 -6
- 3-27: Compact subspace of the real line.
- Cantor's nested intervals theorem.
- Heine-Borel theorem.
- Subset of R^n is compact iff closed and bounded.
- Extreme value theorem.
- 3-27: Lebesgue number lemma.
- Theorem 27.6: Uniform Continuity Theorem.
- Week 15: December 9 -13
- TBA
- Exam review.
- Exam III: Friday, December 13. 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