Topology I
This is a course intended for first year M.Math students.
The following syllabus (not strictly same as the official syllabus) is only indicative of the topics to be covered. The topics covered so-far in the class have been coloured in green.
Syllabus: Metric spaces, Examples, Completion of metric spaces. Baire Category Theorem and applications, sequential compactness, Arzela-Ascoli theorem.Topological spaces, Subspace topology, Hausdorff spaces. Continuous maps: properties and constructions; Pasting Lemma. Homeomorphisms. Product topology, Connected and path-connected spaces. Countability and separation axioms, Compact spaces. Locally compact spaces, one-point compactification, Urysohns lemma, Tietze extension theorem, Urysohn's metrization theorem and partitions of unity. Tychnoff's theorem and Stone-Cech compactification. Quotient topology.Fundamental groups, Homotopy equivalence, Deformation retracts, Seifert-Van Kampen theorem, Covering spaces.
Interesting Topics Not Covered : Nets and Filters (see links below), Paracompactness, Nagata-Smirnov metrization theorem (See Munkres).
Teaching Assistant : Mohan R. Office : S-4 Office Hours : Mondays, 16.15-17.15. (Changes will be informed via email) Email : rajmohan_rs [AT] isibang.ac.in
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.
5. L. Steen and J. A. Seebach : Counterexamples in Topology
6. S. Kumaresan : Topology of Metric spaces.
Assignments : #1 , #2, #3, #4, #5, #6, #7, #8
For past exams, see the following webpage and for time-table, exam schedule et al., see the following webpage.
You can also check some assignments and online notes available on my previous course webpage. Any other necessary additional material will be uploaded / linked on this page as and when needed.
Additional Notes : (unless mentioned the notes are not written by me)
1) Some of the blogs from this page (Especially Notes 7-11 can be very useful for this course and further reading).
2) Completion of metric spaces via Cauchy Sequences
3) A bit on history of the subject and some more.
4) Minimal Uncountable Well-ordered set.
5) Axiom of Choice Equivalents.
6) Filters (David MacIver), Nets and Filters (Philip Parker).
7) 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).
8) Applications of Arzela-Ascoli Theorem from P 151-156 of Goffman, C. : Preliminaries to Functional Analysis in "Studies in Modern Analysis".
Grading :
Assignments / Quizzes : 25
Mid-Semester : 25.
End-Semester : 50
Total : 100.