About the course:

The course will start with definition of metric spaces and topological spaces and proceeds to study topological aspects of metric spaces. W e prove three very important theorems on complete metric spaces and give construction of completion of metric spaces. The course then takes up the study of topological spaces, constructing new topologies from the old. Bases and subbases, I and II countability, separability, connected and path connectedness, Compactness , Lindeloffness, separation axioms etc. The course ends with the celebrated results such as Urysohn’s lemma and Titze’s extension theorem and some applications. The content of this course is mandatory for any meaningful study of analysis and further topology. The lecture slides are backed up by full notes, a strong team of tutors who will handle all the queries sympathetically and also by a number of online interactive session.

The course will start with definition of metric spaces and topological spaces and proceeds to study topological aspects of metric spaces. W e prove three very important theorems on complete metric spaces and give construction of completion of metric spaces. The course then takes up the study of topological spaces, constructing new topologies from the old. Bases and subbases, I and II countability, separability, connected and path connectedness, Compactness , Lindeloffness, separation axioms etc. The course ends with the celebrated results such as Urysohn’s lemma and Titze’s extension theorem and some applications. The content of this course is mandatory for any meaningful study of analysis and further topology. The lecture slides are backed up by full notes, a strong team of tutors who will handle all the queries sympathetically and also by a number of online interactive session.

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.

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

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 April 2023** 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

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

Date and Time of Exams:

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.

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

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

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