SYLLABUS


MATH 461 Topology Spring, 2015



Instructor: Ivan C. Sterling, 177 SH, (240-431-8185), isterling@smcm.edu.


Office Hours: TR 12-2, and by appointment


Book: Topology, 2nd Edition, James R. Munkres


You must attend all classes and be on time. You must work on topology while in class.


Core Topology: Chapter 2: §§12-21 (7 Exercise Sets). Chapter 3: §§23,24,26,27,28,29 (6 Exercise Sets). Chapter 4: §§30-34 (5 Exercise Sets).


The course will be self-paced. There is a wide range of backgrounds, abilities, interests and sedulities among the enrolled students. For some students this is their first upper division course in mathematics and for others the Core Topology will be straighforward and easy. My goal is to optimize the learning experience of everyone. Anyone who knows union and intersection can do topology! You'll need to read the book, discuss with other students and me, learn the material, and solve some problems from each of the Eighteen Exercise Sets. Whenever you feel prepared you can take a pass/fail quiz on an Exercise Set. If you fail a quiz, you can retake it. The quizzes are not meant to be stressful, but only to check that you understand the material. Explaining to me your work on an Exercise Set should typically suffice to pass a quiz. Knowing the vocabulary and basic examples will be important!


Grading: Pass 15 quizzes → D; 16 quizzes → C-, 17 quizzes C; 18 quizzes C+. My expectation is that everyone will finish all their quizzes long before finals, but in any case all quizzes must be completed by Thursday, May 7th, 12 noon.


Projects: If you pass 18 quizzes, then you can receive a better grade (B-,B,B+,A-,A) by doing one or more projects, with or without a partner. Some of the projects will be selected for presentation to the class, either during a class period or at our final. Our final is Thursday, May 7th, 9-11:15 am.


Possible Projects from the Book: The starred(*) Sections from Chapters 2-4. Topics from Chapters 5-8, especially Tychonoff's Theorem. Topics from Algebraic Topology, especially Chapter 9. (Note that Chapters 5-9 are independent from one another.)


Possible Projects outside the Book: Differential Topology, Knot Theory, Homotopy Type Theory, Something else you might be interested in.


Possible Computer Topology Projects: For ideas and for fun: Go to the “Demonstration Wolfram” website and put in the search word 'topology'.





Miscellanea:


  1. Some material overlaps with Analysis I & II. For example §27 “Compact Subspaces of the Real Line”, is similar (but different!) than material covered in Analysis I.

  2. Munkres, p45, defines a countable set to be one that is either finite or countably infinite. Some books do not consider finite sets to be countable.

  3. Our course starts with §12, by the end of §14 more than ten topologies on the reals will be given.

  4. According to Munkres there is only one “deep theorem” in my Core Topology list: §33 p207, Urysohn's Lemma. The second deep theorem, not part of my Core Topology, but a possible Project: §37 p250, Tychonoff's Theorem.

  5. Derivatives are never mentioned in Munkres.

  6. A word about axioms. Most of this is not relevant for this course in particular, but might help students sort out how mathematics works!


Axioms of Mathematics – Logic (not, and, or, implies, if and only if, for all, there exist), plus Eight basic Axioms of Math, plus the Axiom of Choice (§9). It is known, and in this class we assume that the Axiom of Choice is equivalent to the Well-ordering Theorem (§10), the Maximum Principle (§11), and Zorn's Lemma (§11). Munkres does not use Choice or Well-Ordering until Chapter 3. He doesn't use the Maximum Principle or Zorn's Lemma until Chapter 5, which is not part of our Core Topology.


Axioms of the Reals – The reals are defined (and constructed) to be the unique complete ordered field with the Axiom of Completion (which Munkres calls the Least Upper Bound Property). In Analysis it is shown that the Axiom of Completion is equivalent to several other “Axions”*.


Axioms of Topology – §30 “The Countability Axioms”, §31 “The Separation Axioms” … There are many of these type of axioms, which are various extra conditions on a space X. Example: If X satisfies the Hausdorff axiom, then blah blah must be true.



*The Nested Interval Property (every nested sequence of closed intervals has nonempty intersection), the Monotone Convergence Theorem (every monotone bounded sequence is convergent), Bolzono-Weierstrauss (every bounded sequence has a convergent subsequence), and Cauchy's Condition (every convergent sequence is Cauchy).