Course Name	Topology I
Credits	3
Professor(s)	Name	Young Bae Jun
	Department/School	Department of Mathematics Education

COURSE DESCRIPTION

Course Objectives

General topology, also called point set topology, has recently become an essential part of the mathematical background of both graduate and undergraduate students. Based on the basic concepts which are covered in the second grade, this lecture contains what I consider to be the bare essentials every student should know before beginning the study of Separation Axioms, Compactness, Product Spaces, and Connectedness.  Lectures covered include the basic properties of Topological Spaces, Bases and Subbases, Continuity and Topological Equivalence, Metric and Normed spaces, and countability. Each lecture begins with clear and pertinent explanation on basic definitions, principles and theorems together with illustrative and other descriptive material.

MAIN TEXTBOOK and FURTHER READINGS

Main Textbook:

- S. Lipschutz, Theory and Problems of General Topology,   McGraw-Hill Book. Co.,  (1965).

Further Reading:

-S. W. Davis, Topology, McGraw-Hill Book. Co., (2005).

-R. H. Kasriel, Undergradulate Topology, W. B. Saunders Company, London (1971).

TASKS AND GRADING

Tasks- To solve Exercises in each subject (Solving problems and submitting reports)

- To make a new problem from old one (Making and solving new problems). Grading Scheme- Attendance and Tasks : 10%

- First Exam : 20%

- Second Exam (Mid-term Exam) : 25%

- Third Exam  : 20%

- Fourth Exam (Final Exam) : 25%

COURSE SCHEDULE

Course OutlineWeek 1 : Topology of the line and plane.

Week 2 : Topological spaces, Accumulations points, Closed sets, Closure of a set.

Week 3 : Interior, Exterior, Boundary, Neighborhood (systems), Convergent sequences.

Week 4 : Subspaces, Relative toplogies, Equivalent definitions of topologies.

Week 5 : Base for a topology, Subbases.

Week 6 : Topologies generated by classes of sets.

Week 7 : Local bases.

Week 8 : Continuous functions (and arbitrary closeness), Continuity at a point.

Week 9 :  Sequential continuity at a point, Open and closed functions.

Week 10 : Homeomorphic spaces, Topological properties.

Week 11 : Topologies induced by functions, Metrics.

Week 12 : Distance between sets, Open spheres, Metric topologies.

Week 13 : Properties of metric topologies, Equivalent metrics, Metrization problems.

Week 14 : Hilbert spaces, Convergence and continuity in metric spaces,

Normed spaces.

Week 15 : First countable spaces, Second countable spaces, Separable spaces.

CONTACT DETAILS

Tel.: +82 55 751 5674

e-mail :

ybjun@gnu.ac.kr