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An invitation to Topology

By Prof. Indrava Roy   |   IMSC
Learners enrolled: 348
Topology is the study of abstract shapes and their properties which do not change under stretching or squeezing an object without tearing. In this course, we shall investigate and formalize the abstract notion of a “space” under a bare minimum set-theoretic structure, called a topology. This basic structure gives rise to a beautiful and rich source of ideas which at the same time greatly simplify and extend familiar objects in mathematics, such as limits and continuous functions. Arguably, topology (along with the theory of numbers) lies at the heart of mathematics. I invite you to explore this abstract yet foundational theory which plays a significant role in modern mathematics, and by extension, other mathematical sciences.

PRE-REQUISITES: Nill

INTENDED-AUDIENCE: Nill

INDUSTRY-SUPPORT: Nill
Summary
Course Status : Upcoming
Course Type : Core
Duration : 12 weeks
Start Date : 24 Jan 2022
End Date : 15 Apr 2022
Exam Date : 24 Apr 2022 IST
Category :
  • Mathematics
Credit Points : 3
Level : Undergraduate



Course layout

Week 1: Set theory and Logic: Sets and functions, Finite and Infinite sets, Well-ordered sets, Zorn’s lemma and the Axiom of Choice
Week 2: Topological spaces: the basic axioms of topology with examples, Bases and Subbases, Various kinds of topologies on the real line
Week 3: Limit points and closed sets, continuous functions
Week 4: Product topology, Quotient topology and Metric topology
Week 5: Connectedness: connected spaces and subspaces, Connectedness of the real line, Intermediate value theorem
Week 6: Connected components, Path connected, locally connected, and locally path-connected spaces
Week 7: Compact spaces: open cover characterization, Finite intersection property, various notions of compactness 
Week 8: Compact subspaces of the real line, Heine-Borel theorem, extreme value theorem
Week 9: Urysohn’s Lemma and Tietze extension theorem on metric spaces
Week 10: Complete metric spaces, Totally bounded metric spaces and compactness, Lebesgue number lemma
Week 11: Function spaces and the Arzela-Ascoli theorem
Week 12: Baire spaces and the Baire category theorem 

Books and references

Topology, by James R. Munkres

Instructor bio

Prof. Indrava Roy

IMSC
Indrava Roy is an Assistant Professor of Mathematics at the Institute of Mathematical Sciences, Chennai

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: 24 April 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 Madras. 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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