# Math 310 - Topology

[Top]     [Home]             Instructor: Alex Degtyarev     Office: SA 130     Phone: x2135     Mail: no spam
Monday  15:40-17:30    Thursday  15:40-16:30                Office hours:     Monday  14:40-15:30    Thursday  14:40-15:30
 Midterm I Spring 1997 Spring 1998 Spring 1999 Spring 2000 Midterm II Spring 1997 Spring 1998 Spring 1999 Spring 2000 Final Spring 1997 Spring 1998 Spring 1999 Spring 2000

 Class syllabus Detailed weekly schedule Topics causing problems Exam rules and terms Class roster

Textbook:     James R. Munkres, Topology: A First Course. (Prentice-Hall, NY, 1975)
Supplementary:     D. B. Fuks, V. A. Rokhlin, Beginner's Course in Topology. (Springer-Verlag, 1984)
Tentative course contents
• Metric spaces; notion of continuity; open and closed sets.
• Topological spaces: fundamental properties, continuous maps.
• Sequential vs. topological definitions.
• Topological constructions: subspaces, sums, products, quotient spaces.
• More subtle topological properties: connectedness and path connectedness; connected components; countability axioms; compactness and sequential compactness; compactification; separation axioms.
• Urysohn lemma; Tietze extension theorem; metrizability.
• Notion of homotopy and homotopy equivalence.
• Path homotopies; the fundamental group; applications.
• Covering spaces.
• The fundamental group of a CW-complex (if time permits).
• Compact surfaces (if time permits).
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Midterm I     (25%)   October 14, 2005, 1:40 pm     See   important remarks

Topics covered (tentative):
• Metric spaces; continuity; open and closed sets
• Topological spaces; basic properties; closure, interior
• Continuous maps (the concept and basic properties)
• Topological constructions: subspace, product, sum, quotient
• The concept of "universal object"

Chapters:     2
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Midterm II     (25%)   December 9, 2005, 1:40 pm     See   important remarks

Topics covered (tentative):
• The countability axioms (including separability and Lindelöf), simplest relations, behavior under constructions (sub-/quotient spaces, sums, products)
• The separation axioms: simplest relations, behavior under constructions (sub-/quotient spaces, sums, products)
• The Urysohn lemma; the Tietze extension theorem; the Urysohn metrization theorem
• Connectedness and path connectedness; simplest properties and relations; behavior under constructions
• Connected subsets of the real line; the intermediate value theorem
• Components, quasi-components, and path components; local (path) connectedness
• Components as topological invariants; applications
• Compactness; simplest properties; relations to other axioms; behavior under constructions
• Compactness vs. sequential compactness
• Compact subsets of Euclidean spaces; the Weierstraß theorems
• Local compactness and one point compactification

Chapters:     3, 4 (except 4.5)
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Final     (40%)   December 25, 2005, 15:30 (SBZ03)     See   important remarks

Topics covered* (tentative):
• Homotopies and their properties; homotopic maps
• Homotopy equivalence of topological spaces; basic examples
• The fundamental group and the induced homomorphisms
• Applications of the fundamental group
• Covering spaces and their classification
• The fundamental group of the circle

Chapters:     8 (1--6, 9, 11, 14)
All previous material is fully included!
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Homeworks     (10%)   Approximately weekly
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Remarks

During the exams please keep in mind the following:
• Calculators are  not  allowed
• Identical solutions (especially identically wrong ones) will  not  get credit. I reserve the right to decide what "identical" means. You still have the right to complain
• Do not argue about the distribution of the credits among different parts of a problem. I only accept complaints concerning my misunderstanding/misreadung your solution
• Show all your work. Correct answers without sufficient explanation might  not  get full credit
• Indicate clearly and unambiguously your final result. In proofs, state explicitly each claim
• Do not misread the questions or skip parts thereof. If you did, do not complain
• If you believe that a problem is misstated, do not try to solve it; explain your point of view instead
• Each problem has a reasonably short solution. If your calculation goes completely out of hands, something must be wrong