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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    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

    Grading policy
    I will take off a few (2-3) points for arithmetical mistakes. However,  a lot  of points will be taken off for `obvious' mistakes, i.e., either those that you can easily avoid or those showing a deep misunderstanding of the subject.
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