An introduction to Point-Set-Topology Part-II

By Prof. Anant R. Shastri | IIT Bombay

Learners enrolled: 445

About the course: This course begin with the assumption that the audience has attended the part-I. We cover all important advances topics such as Various notions of compactness in metric spaces, Ascolii’s theorem, paracompactness, compactifications, Urysohns Metrization theorem, Stone-Weiertrass theorem, totally disconnectedness, etc. Checking whether any of these topological properties in product invariant occupies a considerable amount of our time. A brief introduction to Topological Dimension Theory, discussion of function-space topologies and finally the Ordinal topology as a rich source of counter examples are the added features. The content of this course will be useful for any body who wishes to study deeper aspects of Analysis or topology or apply topological tools anywhere else in Mathematics or Physics, etc. It is certainly useful in the two course that I have offered on Algebraic Topology on NPTEL portal.

PRE-REQUISITES: Attended the part-I of this course from NPTEL or acquired the basic knowledge the content offers from elsewhere.

INTENDED AUDIENCE: Students from various streams which include some modern mathematics such as B. Sc., M. Sc., Students, Ph. D. , B. Tech. and M. Tech. who have seriously attended PST-I or have acquired the basic knowledge the content offers from elsewhere, will be able to benefit from this course.

Summary

Course Status :	Completed
Course Type :	Elective
Duration :	12 weeks
Start Date :	25 Jul 2022
End Date :	14 Oct 2022
Exam Date :	30 Oct 2022 IST
Enrollment Ends :	08 Aug 2022
Category :	Mathematics
Credit Points :	3
Level :	Undergraduate/Postgraduate

Page Visits

Course layout

Week 1: Compactness and separation axioms.

Week 2: Paracompactness and partition of unity.

Week 3: Various notions of Compactness in metric metric spaces, Ascoli’s theorem.

Week 4: Productive properties.

Week 5: Productive properties continued.

Week 6: Urysohn’s Metrization theorem.

Week 7: Compactifications 1-pt compactification, Stone-Cech Compactification.

Week 8: Totally disconnectedness.

Week 9: A brief introduction to dimension theory.

Week 10: Function spaces, Compact open topology, exponential correspondence.

Week 11: Stone-Weierstass Theorem.

Week 12: Ordinal Topology- a source for many counter examples.

Books and references

There are many good books available on this topic though no single one will do for this course. Therefore, full notes will be made available to all enrolled students which will be the text-book for this course. Set of further references will be given on the very first day and will be also mentioned in these notes.

Instructor bio

Prof. Anant R. Shastri

IIT Bombay

Prof. Anant R Shastri is a retired Emeritus Fellow of Department of Mathematics I.I. T. Bombay. After serving in School of Mathematics T.I.F.R. for 16 years I joined I.I.T. Bombay as a full professor in 1988. Apart from several research papers,I have published three books. Since 2004, I have constantly involved in the activities of ATM schools, The chief activity of these schools is to impart advanced training in Mathematics to Ph. D. students in various universities and research institutions in the country. These activities were initially funded by NBHM and currently adapted by National Centre for Mathematics, I.I.T. Bombay. Over the years, I have taught t roughly the contents of the proposed course more than 20 times to M.Sc/ BTech / MTech and Ph. D. students.CDEEP has recorded my course on Complex Analysis MA205 for B. Tech students. I have offered a course in Algebraic Topology in two parts on NPTEL portal. I strongly believe that especially in these troubled times NPTEL mode of Knowledge-dissemination is the best

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams: 30 October 2022 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.

Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Bombay .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team