Topology-I

This is a course for first year JRF/Ph.D students.

Syllabus :

Topological Spaces: Topological spaces, Bases, Continuous maps, Subspaces, Quotient spaces, Products, Connectedness and Compactness.

Convergence: Nets, Filters, Limits; Convergence, Countability and Separation axioms.

Topological groups: Topological groups; Uniform structures, Products of Compact spaces; Compactifications.

Metrizability: Metrizability and Paracompactness, Complete Metric spaces and Function spaces.

Monodromy: Fundamental Group and Covering spaces.

References :

1. J. R. Munkres, Topology: a first course.

2. K. Janich, Topology.

3. M.A. Armstrong, Basic Topology.

4. G.F. Simmons, Introduction to Topology and Modern Analysis.

Some Additional Material (Unless mentioned, these are not written by me)

  1. Filters (David MacIver), Nets and Filters (Philip Parker).

  2. Proof of Erdos - De Bruijn Theorem on Chromatic number of Graphs via Tychnoff's theorem : Theorem 8.1.3 of Graph Theory by R. Diestel. For other similar applications, see the following article (Nick Lowery). Application to Invariant means (Ken Brown).

  3. Applications of Arzela-Ascoli Theorem from P 151-156 of Goffman, C. : Preliminaries to Functional Analysis in "Studies in Modern Analysis". This was partly a student presentation.

  4. Stone-Weierstrass Theorem. This was presented in the class by a student based on Chapter 7 of G. F. Simmons.

  5. Picard's existence and uniqueness Theorem. Student presentation based on Appendix A of G. F. Simmons.

Assignments : #1, #2, #3, #6, #8.

Not all assignments are uploaded.