Topology-I

Syllabus Details

• Unit I: (Review of metric spaces and definition of topological spaces) A brief review of the theory of metric spaces with emphasis on completeness and compactness; topological spaces; subspace topology.
• Unit II: (Basic notions about topological spaces-I) Closure, interior, and boundary of a set; limit points of a set; isolated points; dense set; nowhere dense set; set of first category; set of second category; Baire category theorem.
• Unit III: (Basic notions about topological spaces-II) Continuous functions between topological spaces; base and subbase; projections and the product topology; separable spaces; first countable spaces; second countable spaces.
• Unit IV: (Separation axioms, and compactness) Hausdorff spaces; regular spaces; normal spaces; Urysohn’s metrization theorem; compact spaces and their basic properties; continuous functions and compactness.
• Unit V: (Special types of topological spaces) Locally compact spaces; connected and path connected spaces; totally disconnected.
• Unit VI: (Advanced topics in Topology) Alexander subbase theorem; Tychonoff theorem; quotient maps and the quotient topology; the space of continuous real-valued (or complex-valued) functions on a compact space; Tietze extension theorem.

Course Learning Objective

1. Explain the notion of completeness and compactness of metric spaces and solve elementary problems involving these notions and the basic concepts about topological spaces.
2. Explain the notion of base and subbase, product topology, separation axioms, countability axioms, and solve elementary problems involving these notions.
3. Explain the notion of compactness, connectedness, quotient topology, and spaces of continuous functions, and solve elementary problems involving these notions.
4. Analyze the proofs of results in Topology with a critical mind and use those results and the proof-techniques to analyze and decide, with suitable justification, whether a given statement in Topology is true or false.
5. Relate the theory and theoretical structure of Topology as a prerequisite for the future study subjects such as Functional Analysis, Algebraic Topology, and Differential Topology.

Recommended Books

1. Dugundji, James, Topology, Allyn and Bacon Series in Advanced Mathematics, Allyn and Bacon, Inc., Boston, Mass.-London-Sydney, 1978.
2. Munkres, James R, Topology: A First Course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975.
3. Kelley, John L., General Topology, Graduate Texts in Mathematics, No. 27, Springer-Verlag, New York-Berlin, 1975.
4. Willard, Stephen, General topology, Reprint of the 1970 original [Addison-Wesley, Reading], Dover Publications, Inc., Mineola, NY, 2004.