Topology

By Mandar Bhanushe | University of Mumbai

Learners enrolled: 1228

In this course we shall come across important notions like continuity, convergence, compactness, separability, connectedness which are important in many applied areas of Mathematics. We shall do various definitions, theorems and their proofs from topics in Topology. One more objective which unknowingly be achieved is the increase in our power of abstraction in the due course. We shall:

1. Understand standard concepts of Set theory

2. Analyse structure of Sets and other abstract algebraic structures

3. Understand definitions; construct examples and counter examples based on definitions

4. Develop intuition regarding proofs, make arguments based on logic

We shall cover the following topics in this one-semester 3 credits course on Topology:

1. Review of set theory, relations and functions

2. Introduction to topology of metric spaces

3. Introduction to topological spaces

4. Continuity, convergence

5. Subspaces, product spaces, quotient spaces

6. Connectedness and Compactness

Summary

Course Status :	Completed
Course Type :	Core
Duration :	12 weeks
Category :	Mathematics
Credit Points :	3
Level :	Postgraduate
Start Date :	13 Jan 2020
End Date :	25 Apr 2020
Enrollment Ends :	08 Mar 2020
Exam Date :

Page Visits

Course layout

WEEK 1 :

Set Theory, Relations and Functions

WEEK 2 :

Metric spaces

WEEK 3 :

Interior/Boundary/limit points, closure

WEEK 4 :

Sequences, Convergence and properties

WEEK 5 :

Completion, Compactness and Connectedness of Metric Spaces.

WEEK 6 :

Compactness & Equivalent Conditions for Compact Metric Spaces

WEEK 7 :

Topological Spaces, Basis, suspaces, product topology

WEEK 8 :

Separation axioms, First and Second Countability, Baire space

WEEK 9 :

Compactness in general topological spaces, Continuity and compactness,

Subspace, Lebesgue covering lemma.

WEEK 10 :

Tube lemma and it's application, Metrizable spaces and heriditary properties of topological spaces.

WEEK 11 :

Product of Regular spaces, Connectedness.

WEEK 12 :

Quotient topology, quotient spaces and Tychonoff theorem.

Books and references

Topology by James Munkres
Topology of Metric Spaces by S. Kumaresan
Introduction to General Topology by K D Joshi

Instructor bio

Mandar Bhanushe

University of Mumbai

Mandar Bhanushe

Mathematics Faculty at the Institute of Distance and Open Learning,
University of Mumbai.

He has an experience of 15 years teaching both at the undergraduate and postgraduate level.

He was winner of National level Indian Education Award at the Indian Education Congress in May 2013 for conducting Best Webinar Series in the subject of Mathematics.

He has trained more than 1300 College/University teachers in the area of e-learning and Learning Management Systems.

He has authored 4 books in Mathematics at the undergraduate level.

He is the founder Chairman of Raising a Mathematician Foundation which works in the area of Mathematics Education for school students and has conducted workshops/camps for high school students, teachers and parents.

He has presented papers at various National/International Conferences.

He has coordinated two Refresher Courses and One Short Term Course organised by University of Mumbai's UGC-HRDC.

He has worked as a subject expert for training high school mathematics teachers of Maharashtra along with IIT-Bombay and RMSA.

He is also a member of a NCERT Committee for analyzing the fear of Mathematics and put forth the recommendations to address the problem.

Course certificate

The course is free for learners to enroll, learn and participate in the discussion forums and attempt quizzes. For getting completion certificate, learner has to register for the examination and write the proctored exam conducted by SWAYAM in person at any of the designated exam centers near him/her.

• The exam fee is of Rs 1000/- (as of now).

• Details of Examination:

09:00 - 12:00 Noon Morning Session - 09th May 2020

• 30 Marks will be allocated for Internal Assessment and 70 Marks will be allocated for Proctored Examination.

Total Marks: 100 ; Passing Marks: 40