MA 231: Topology

August 2021

Syllabus

Open and closed sets, continuous functions, Metric topology, Product topology, Connectedness and path-connectedness, Compactness, Countability axioms, Separation axioms, Complete metric spaces, Quotient topology, Topological groups, Orbit spaces, Urysohn’s lemma, Metrizability, Baire Category theorem.

Suggested books:

1. Armstrong, M. A., Basic Topology, Springer (India), 2004.
2. Munkres, J. R., Topology, Pearson Education, 2005.
3. Viro, O.Ya., Ivanov, O.A., Netsvetaev, N., and Kharlamov, V.M., Elementary Topology: Problem Textbook, AMS, 2008.

Assignments

Assignments will be posted roughly once a week.

1. Topological Spaces due by Monday, Aug 16, 2021.
2. Bases and Metrics due by Monday, Aug 23, 2021.
3. Subsets and Metrics due by Monday, Aug 30, 2021.
4. Continuity due by Monday, Sep 13, 2021.
5. More Topologies and Homeomorphisms due by Monday, Sep 20, 2021.
6. Connectedness and Applications due by Monday, Oct 11, 2021.
7. Separation Properties due by Monday, Oct 18, 2021.
8. Countability and Compactness due by Monday, Oct 25, 2021.
9. Compactness for Metrics and Compactification due by Monday, Nov 1, 2021.
10. Products and Metrization due by Monday, Nov 15, 2021.
11. Baire Theorem and Quotients due by Monday, Nov 22, 2021.

Course Details

• Instructor: Siddhartha Gadgil
• E-mail: siddhartha.gadgil@gmail.com
• Teaching Assistants: Pranab Sarkar, Pabitra Barman

This course will be taught online, mostly through pre-recorded lectures. We will mostly follow the book Elementary Topology: Problem Textbook by Viro et. al., which is available online. There will also be interactive sessions on Microsoft Teams.

Microsoft Teams details

A Team for this course (within IISc) named Topology : August 2021 has been created. A link to join this Team will be posted on the IISc intranet (under Department of Mathematics), and people from IISc can also join using the Team code nq02o79. Please join the Team if you register for the course or wish to attend interactive sessions. If you choose to leave the course after some lectures you can leave the Team.

Please download the appropriate Teams app before the first interactive session. There are Apps available for Windows, Linux (Debian/Ubuntu and Redhat), Mac, Android and iOS. Once you have joined the team, whenever I (or someone else) starts a meeting you will see in the chat that a meeting has started. Please click to join.

The interactive sessions will be weekly on Mondays and Wednesdays, 9:00 am to 10:00 am, starting on Wednesday, August 4, 2021.

I would encourage using the Teams chat when it is set up to ask questions whenever needed. We will also meet for interactive sessions.

Assignment submissions

Assignments will be posted on this web site and linked in Microsoft Teams. They have to be submitted through Microsoft Teams. The easiest way to do this is to write on paper, photograph or scan this and submit pdf or image files. It is recommended that you use an app like Adobe scan or Microsoft Lens to generate a single pdf file for each assignment (instead of a collection of images).

If you wish you can instead use LaTeX and submit pdf files.

Examinations and Grades

Due to the difficulty of online evaluation, there will be 3 term examinations as well as a final examination. These will all be written, followed by an oral section where the student must be able to explain anything in the written proof, well enough to establish to my satisfaction that the student wrote this based on understanding, not by copying. Suspicion of copying will be strongly penalized.

Grades will be based on the assignments, midterm examinations and final examinations with the following weightages.

• Assignments: 10%
• Midterm 1: 15%
• Midterm 2: 25%
• Midterm 3: 20%
• Final exam: 30%

Midterm I

The following are the details for the midterm I.

• Date: Saturday, September 4, 2021.
  • Written Exam: 9:00 am to 10:00 am.
  • Oral follow-up: Short sessions with individual students between 10:00 am and 12 noon.
• Syllabus: Topological spaces; Bases; Metric spaces; Subspaces; Subsets and points, interiors and closures (up to Lecture 6).
• Weightage: 15% of the total.

Midterm II

The following are the details for the midterm II.

• Date: Saturday, October 2, 2021.
  • Written Exam: 8:30 am to 10:00 am.
  • Oral follow-up: Short sessions with individual students between 10:00 am and 12 noon.
• Syllabus: The contents up to lecture 11. Details of the contents are on the lectures page.
• Weightage: 25% of the total.

Midterm III

The following are the details for the midterm III.

• Date: Friday, November 12, 2021.
  • Written Exam: 9:00 am to 9:50 am.
  • Oral follow-up: There will be no oral component.
• Syllabus: The contents up to lecture 19. Details of the contents are on the lectures page.
• Weightage: 20% of the total.

Final Examination

This will be a written examination only.

• Date: Saturday, December 4, 2021.
• Time: 9:00 am to 12 noon (written exam only)
• Syllabus The contents of all lectures. Details of the contents of the lectures are on the lectures page.
• Weightage: 30% of the total.