New Course Proposal for Math 4381/6381,

General Topology

Statement. Topological spaces show up naturally in mathematical analysis, abstract algebra and geometry. This has made topology one of the great unifying ideas of mathematics. General topology, or point-set topology, defines and studies some useful properties of spaces and maps, such as connectedness, compactness and continuity. The subject of topology is concerned with those features of geometry which remain unchanged after twisting, stretching or other deformations of a geometrical space. The influence of topology is also important in other mathematical disciplines such as dynamical systems, algebraic geometry, differential geometry and certain aspects of combinatorics. Actually, topology has introduced a new geometric language, and is being applied to fields outside of mathematics such as molecular biology, astronomy and physics (the shape of space), computer science, robotics, psychology and linguistics. For example, the importance of topology to biology results from the ability of topological changes to manifest as geometric changes in twist and writhe. DNA topology studies the shape and path of the DNA helix in 3-dimensional space.

Topology is concerned with spatial properties preserved under bi-continuous deformations, that is, stretching without tearing or gluing. Topology also deals with the concept of nearness at various levels. In point-set topology there are nearness relations between points and sets, together with continuous transformations which preserve these nearness relations. In applications, the abstract sets are replaced by sets of relevant elements.

The goal of this course is to provide math majors, as well as students from other disciplines with the necessary prerequisites, with a firm grounding in point-set topology. The approach will be axiomatic, and understanding theorems and their proofs is essential, but the course will also have a component which is focused on building intuition and developing the “big picture”. It is expected that, by the end, the students will be able to solve problems using the definitions and theorems of general topology, and that they will possess a clear idea of how topology relates and can be applied to other areas, within and outside of mathematics.

It is very important to note that from about 1925 to 1975 Topology was considered the most important growth area within mathematics. What’s more, much of the development in other mathematical areas would not have occurred without the tools provided by Topology. A Topology course gives a mathematics student the option of acquiring a more well-rounded profile.

Currently our GSU mathematical curriculum does not include any course in which topology is the focus of study. The proposed course will help to overcome this drawback and to achieve the following important goals:

Ø  Contribute to our undergraduate and graduate program in Mathematics.

Ø  Motivate our students and give them a strong background in a fundamental area of Mathematics, which will permit future success in graduate studies for those that decide to pursue them.

Placement of the New Course in the GSU Curriculum.

·  Graduate Catalog: Master of Science in Mathematics ‘Additional graduate – level courses in mathematics, computer science, or a related field’.

·  Undergraduate Catalog: B.S. in Mathematics, Area G(2) ‘Mathematics Electives’.

Textbook.

1.  Munkres, James, 2000, Topology, Second Edition. Prentice Hall, Inc., Englewood Cliffs: N.J.

Offering Schedule.

The course will be taught once a year.

Staffing.

M. Montiel

F. Enescu

A, Smirnova

Z. Li

______

PROPOSED SYLLABUS

General Topology

4381/6381

Instructor: Mariana Montiel

Office: 708-College of Education

Phone: (404) 651-0646

E-mail:

Prerequisites: Math 4441/6441

Topics to be covered: Review of aspects of set theory and logic; topological spaces and continuous functions (basis for a topology, product topology, order topology, subspace topology, metric topology, quotient topology, homeomorphisms), connectedness and compactness in topological spaces, countability and separation axioms, the fundamental group (homotopy of paths, covering spaces, the fundamental group of the circle, retractions and fixed points).

Textbook: Munkres, James, 2000, Topology, Second Edition. Prentice Hall, Inc., Englewood Cliffs: N.J.

Alternative Textbook:

Related reading:

·  Armstrong, M.A., 1983, Basic Topology, Springer Science+Business Media, New York: USA.

·  Crossley, Martin D. 2005, Essential Topology, Springer-Verlag London Limited.

·  McCluskey, Aising & McMaster, Brian, 1987 Topology Course Lecture Notes: http://at.yorku.ca/i/a/a/b/23.htm

·  Weeks, Jeffrey T., 1985, The Shape of Space, Marcel Dekker, INC, New York: USA.

Grading:

Grading will be based on:

1)  3 exams and a final

2)  Class presentations of the assigned material (at least two per person per semester).

3)  A Final Project, which will be oriented towards some aspect of application of topology to other areas of mathematics or other disciplines.

Graduate students will be expected to answer some extra problems on the exams, and to present a less general and more content focused project.

The exams will be 45% of the final grade, the final exam will be 20% of the final grade, the class presentations will be 15% of the final grade and the project will be 20% of the final grade.

The grading scale, according to Departamental policy is:

A: 90-100

B+: 87-89

B: 80-86

C+: 77-79

C: 70-76

D: 67-69

D: 60-66