- A Boring Introduction You Can Skip
- Open Intervals and Open Sets in Real Line
- A Special Type of Open Sets
- Reformulating Definition of Open Sets.
- Some Interesting Properties of Open Sets.
- (Optional) Generalising the Definition of Open Sets
- (Optional) A vague introduction to the Concept of Generalisation in Mathematics.
Introduction to the Coursework
Who is this Course Designed for?
The entire course is compiled for the wonderful PG 2020-22 batch of St. Joseph’s College, Bangalore, whom I taught the same course. I promised them something that will help them review the topics in future, at any point of their lives if needed!
I have tried to assure that it’s as general as possible.
Though I had postgraduate Mathematics majors in my mind when I designed it, I have tried to cover all bases such that it only demands you to be familiar with the concept of sets, set of real numbers (numbers like 0, 1, 2, 0.2, 0.5, 1.6, Square root 2,… ).
In other words, even a school students versed with those concepts should be able to follow this article investing some effort.
The proofs will not be given much focus – the key focus will rest on ideas, application, presenting a new way to learn pure papers. But I’ll try to include each proof for the sake of completeness.
What can you Expect?
I’m someone simply passionate about the subject, and my knowledge is limited. I’m not a researcher in the field or have any credentials in the same. My only experience is in teaching this course twice, and students seemed to be convinced with the topics. So, this coursework will help you build a general outline of the subject and/or help you develop the initial intuitive understanding of the topics. For advanced exploration, you are always suggested to re-explore the topics with a researcher in the field.
Self-paced learning is the tomorrow, as I now see. So, hopefully these articles helps someone in future.
I have also tried to incorporate as many Objective Type questions as possible, with those quickly revising for entrance examinations in mind.
I have also presented an intuitive approach to introduce the subject to a student already familiar with concepts of open sets on real line/metric spaces. Though it serves as a convincing reason to connect what they know and what they are going to learn, (something is better than nothing) I must acknowledge that my expertise in the field is limited. Feel free to present your case for/against it, will greatly help me to improve myself.
The narrative is aimed at encouraging students to think about how the concept might have evolved, and tries to present a structured approach to study the subject.
Note: This is the draft zero of this article – hopefully, if time permits, I’ll keep updating it. You can give your suggestions, if any!
Look for page numbers below!