Math 310 - Topology
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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
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
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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