Department of Mathematics

Department of Mathematics

GC University, Lahore

Course Title:

Topology (16 Weeks)

Course Code:

Math-303Course Length:

Semester

Credits:

3 hrs

Course description: The objectives of Introduction to Topology does not stress the applications of mathematics or the impact of technology, but it does stress the ideas of abstraction, aesthetics, the development of mathematical tools and the use of the language of mathematics.Course Contents: Metric spaces, Holders and Minkowski inequalities, Convergence and Completeness, lp-spaces, Continuous functions, Topological spaces, Open and closed sets, Neighborhoods, interior and exterior of a subset, Limit points, Derived sets, Closure, Subspaces, Bases and sub-bases, First countable spaces and second countable spaces, separable spaces, Filters, Continuous functions, Hemeomorphism, Product spaces, Separation axioms, Baires category theorem, Completely regular spaces, Normal spaces, Connected spaces, Compact spaces, Sequential compactness, Totally bounded sets, Compactness in metric spaces.Prerequisite Course(s): None

Course goals and performance objectives:

Goal I:

Metric and Topological Spaces

(20% of Scheduled Learning Activities)

Obj. 01:Sketch a block diagram of basic ingredients of the subject comprising of

(a) Metric spaces (b) topology and topological spaces

(c) Separable spaces, Connected Spaces(d) Open mappings, continuous functions etc.

Obj. 02:Explain briefly the significance of the above topics.Goal II:Convergence of sequences and Continuous Functions.

(25% of scheduled learning activities)

Obj. 01:Ability to understand open sets, closed sets, ideas of neighborhoods, limit points, closure of a set in metric and topological spaces.

Obj. 02:Knowledge of convergence of sequences, in metric and topological spaces, completeness of space.Obj. 03:Proofs of Holder's and Minkowski inequalities.

Obj. 04:Study of continuous functions and isometric.Goal III:Separation Axioms and Connectedness.

(30% of scheduled learning activities)

Obj. 01:Ability to use the topological concepts to study the sets and spaces.

Obj. 02:Ability to understand the separation axioms.Obj. 03:Ability to explore connected and compact spaces. Goal IV:Compactness.

(25% of scheduled learning activities)

Obj. 01:Ability to find Sequential compactness. Compactness in metric spaces. Obj. 02:Ability to study Totally bounded sets.

Obj. 03:Proof of Baires category theoremTeaching Learning Strategies:

The course aims that upon completion of this course, students should:

be familiar with basic concepts of topology,

gain mathematical maturity,

become competent in writing proofs,

apply spacial imagination to theory.

Time Management Matrix:

Goal 01:

20% - 08 Lectures

Goal 02:

25% - 10 Lectures

Goal 03:

30% - 12 Lectures

Goal 04:

25% - 10 Lectures

Total:

100% - 40 Lectures

Assessment Strategies:

The assessment for curse in a semester will be carried out on the basis of following criteria:

Semester Work

40% - 40 Marks

Final Examination 60% - 60 Marks

Total:

100% - 100 Marks

Semester work includes Quizzes/ Term papers/ projects given by the instructor to the students.

Guidance for term paper:

The students are required to submit term paper in time. The term paper should not merely be descriptive, must include the backgrounds, motivation, application and related problems. Term papers will be assessed on the basis of ones analytical, logical thinking and comprehension of the subject. Web based material should be discouraged.

Recommended Texts:

1. Introduction to Topology by J.V. Deshpende, Tata MeGraw Hill Publishing

Company, 1990

2. Topology, a First Course by J. Munkers, Prentice Hall, 1991.

3. General Topology by S. Willord, Addison-Wesley Publishing Company,

Reading Mess, 1970.

4. Point Set Topology by Steven A. Gaal, Academic Press, New York and London, 1964.

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