ISL Topology (2013)

 Central Theme:– Topology as the study of Convergence and Continuity, with emphasis on Examples and Exercises.

• Confirmed speakers/Instructors

(i) B. V. Rao , (Adjunct Professor), CMI, Chennai
(ii) S. M. Srivatava , ISI, Kolkata
(iii) H. Sarbhadhikari , ISI, Kolkata
(iv) Sanjay K. Pant , DDU College, DU and CEMS, Almora
(v) Anant R.Shastri , IIT Bombay

• Special Lectures/Guest Speakers

(i) M S Raghunathan, NCM,IIT Bombay
(ii) Ravi S Kulkarni, BIM and Pune University

• Course Associates

(i) Pawan K. Tamta, Kumaun University, Almora
(ii) Hemlata Pande, Kumaun University, Almora
(iii) Kushal Lalwani, Delhi University
(iv) Gopal Datt, Delhi University

• Detailed speaker-wise Syllabus
  Part-I Speaker: B. V. Rao (BVR) (6 lectures)

1. Major examples: R, Rn, [0, 1], [0, 1]n.
2. Spaces of continuous functions with various notions of convergence. e.g.poitwise convergence, uniform convergence, uniform convergence on compacta (closed and bounded sets), which leads to compact-open topology. [Initially, examples should be taken from usual analysis]
3. The notion of (usual) distance on the above cited spaces, unit ball etc.
4. Can uniform convergence be covered as a usual notion of convergence using distance? As an answer, introduce metric spaces, continuous functions between them and go on to show that uniform convergence and uniform convergence on compacta of Rn can be described as a convergence using a metric.
5. Denseness of Q and Qn and applications, separability of metric space.
6. Can pointwise convergence be covered as a notion of convergence using metric? As an answer, emphasize the need to introduce a notion of continuity in terms of open sets and closed sets. Define topological spaces and show that further axioms need to be introduced to remove some garbages such as discreet and indiscreet spaces etc.
7. Show that every bounded sequence in Rn has a convergent subsequence.Prove Weierstrass’ theorem and related results.

Part-II S. M. Srivastava(SMS) (6 lectures)

8. Introduce sequential compactness.
9. Capture compactness in terms of closed sets and open sets. Heine-Borel with proof.
10. Basic properties of compact sets. Lebesgue covering lemma. Product of compact spaces.
11. Complete Metric Spaces.

Part-III H. Sarbhadhikari(HS) and Sanjay K. Pant (6 lectures)

12. Covering axioms, compactness, Lindel ̈ off.
13. I and II countability, separability.
14. Separation axioms: T0 to T5. Urysohn’s lemma and Tietze extension theorem.
15. Embedding and metrization.

Part-IV Anant R. Shastri(ARS)(6 lectures)

16. Intermediate value theorem with proof (for interval and then foe convex subset of Rn). Extract the definition of connectedness and define connected spaces. Relation with path-connectedness. Equivalence of LUB property and connectedness.
17. Quotient spaces. Separation properties under quotient topology.
18. Playing with the M ̈obius band.
19. Paracompactness and partition of unity.
20. Induced and coinduces topology, compactly generated topology and function spaces.

 Tentative Time - Table

[Abbreviations of names used in the time-table above are explained in the detailed- syllabus
Tn— nth tutorial
Sn—nth special lecture
Tea Breaks—11.00-11.30 and 15.30-16.00
Lunch Break— 13.00 - 14.30
6 October 2013 Sunday— Trip to Binsar/Jageshwar
11 October Friday— Faculty Dinner.