Course Information

Instructor:

John G. Del Greco, Ph.D.

Office:

Sullivan Science Library, Room 024

Office Hours:

Tuesday and Thursday afternoons, 5:00p.m.-6:00p.m.

Telephone:

Office: (773) 508-3530

E-mail Address:

jdg@math.luc.edu

Math 386/411 World Wide Web Site:

http://www.math.luc.edu/~jdg/courses/m386/sum1_98/syllabus.htm

Class Hours and Location:

Tuesday and Thursday evenings, 6:00p.m.-9:00p.m., DH TBA

Prerequisites:

Math 351

Text:

Elements of Point Set Topology by John Baum. The text is available at the Loyola University Bookstore in the Granada Centre.

Topology Links on the Web:

There are numerous topology resources available on the web. Listed below are some of the instructor's favorite links:

• Topolgy Atlas
• Minnesota Geometry Center
• Yahoo Mathematics Node
• American Mathematical Society
• Society for Industrial and Applied Mathematics

Attendance:

Attendance is not required. Students are responsible for all material discussed in class whether choosing to attend class or not. In addition, handouts will be distributed only once. Students not attending class will have to obtain copies of any handouts from fellow classmates. The instructor will recycle any extra handouts immediately after the class in which they were distributed.

Grading:

Final grades will be computed according to the following recipe. There will be a midterm examination worth 25% and a final examination worth 35%. The final examination will be comprehensive. In addition, there will be four quizzes each worth 10%. Therefore, the quizzes will be worth 40% of the total grade. Quizzes will be take-home. Quizzes will be distributed on the Thursdays designated below and will be collected on the following Tuesday. Quizzes may contain problems that were assigned as homework. Homework will be assigned from the textbook and should be completed before the beginning of the next class. Homework problems will be announced at the end of class and will also be listed on the web. Homework will not be collected or graded. However, students will have the opportunity to discuss homework problems in class. The midterm examination will be on Thursday June 4. It will have two parts: an in-class part and a take-home part. The in-class part will emphasize definitions and basic concepts. The take-home part will consist of problems requiring more time. The final examination will be on Thursday June 25 from 6:00p.m.-9:00p.m. in our classroom. The final examination cannot be taken early. If a student cannot attend an examination, he or she must call the instructor before the examination to arrange an alternate examination date. (If you cannot reach the instructor directly, leave a message with the departmental secretary.) Failure to do so will result is a 0% for that examination. Final grades will be assigned according to the following scale:

• 90%-100%.........................A
• 87%-89%...........................B+
• 80%-86%...........................B
• 77%-79%...........................C+
• 70%-76%...........................C
• 67%-69%...........................D+
• 57%-66%...........................D
• 0%-56%.............................F

The final average will be computed as follows. If mid is the midterm grade, fin is the final exam grade and quiz is the sum of the four quizzes (each quiz being worth 25 points), then the final average, fin_ave, is given by

fin_ave := .25*mid + .35*fin + .40*quiz.

The number fin_ave will be fitted to the above scale. No exceptions will be made for senior expecting to graduate! The take-home quizzes will be distributed on the following days: Quiz 1.....May 21, Quiz 2.....May 28, Quiz 3.....June 11, Quiz 4.....June 18.

Course Goal:

The goal of the course is to equip the student with those tools and techniques from point set topology that are needed for advanced studies in a number of mathematical disciplines. In particular, the course will prepare students for courses in algebraic and differential topology.

Important Dates:

Friday June 12 is the last day a student may withdraw from a course with a non-penalty grade of W. Tuesday June 23 is the last day of class.

Academic Dishonesty:

Prohibited activity includes cheating on a quiz or exam, using forbidden materials on an exam and helping others on a quiz or exam. A student who violates these rules for the first time will receive a failing grade on the quiz or exam on which the cheating occurred. A second violation will result in an F for the entire course.

Course Schedule

1. Topological Spaces
   1. Sections 0.1-0.6, 2.1: Algebra of sets, indexed families of sets, functions.
   2. Sections 1.1-1.2, 1.8: Neighborhood systems, open sets, comparing topologies.
   3. Sections 1.3-1.5: Limit points, derived set, closure, closed sets.
   4. Section 1.6: Subspaces.
   5. Section 1.7: Limits of sequences, Hausdorff spaces.
   6. Section 1.9-1.10, 2.4: Bases, subbases, product topology, countability.

2. Continuous Functions and Homeomorphisms
   1. Section 2.2: Continuous functions.
   2. Section 2.3: Homeomorphisms.
   3. Section 2.4: More on product topologies.

3. Compact Spaces
   1. Section 3.1: Compactness, finite-intersection property.
   2. Section 3.1: Properties of compact spaces.
   3. Compact sets in the real line.
   4. Section 3.1: Tychonoff Theorem.
   5. Section 3.3: Countable compactness.
   6. Section 3.4: Local compactness, one-point compactification.

4. Connected Spaces
   1. Section 4.1: Separation, connectivity.
   2. Section 4.2: Properties of connected spaces.
   3. Connected sets in the real line.
   4. Section 4.3: Components.
   5. Section 4.4: Local connectivity.
   6. Section 4.5: Path connectivity.

5. Separation Axioms
   1. Section 3.2: T_i spaces, i=0,1,2,3,4.
   2. Section 3.2: Regular and normal spaces.
   3. Section 3.2: Examples and counterexamples.
   4. Section 3.2: Completely normal spaces.

6. Metric Spaces and Metrization Theorems
   1. Section 5.1: Metric spaces, metric topology, metrizability.
   2. Section 5.2: Properties of metric spaces.
   3. Section 5.3: Urysohn's Lemma, completely regular spaces, Tietze Extension Theorem.
   4. Section 5.3: Urysohn Metrization Theorem.
   5. Section 5.4: Complete metric spaces.
   6. Section 5.5: Catagory theorems.

7. Introduction to Algebraic Topology
   1. Homotopy of paths.
   2. Fundamental group.

Homework Problems