Topology

By Mandar Bhanushe | University of Mumbai

Learners enrolled: 2253

In this course we shall come across important notions like continuity, convergence, compactness, separabaility, 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. Analyze 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 4 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 :	15 weeks
Category :	Mathematics
Credit Points :	4
Level :	Postgraduate
Start Date :	15 Jul 2019
End Date :	30 Oct 2019
Enrollment Ends :	11 Sep 2019
Exam Date :	10 Nov 2019 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

WEEK 1 :

Set Theory, Relations and Functions

WEEK 2 :

Metric spaces

WEEK 3 :
Interior/Boundary/limit points, open sets, closure

WEEK 4 :

Sequences, Convergence and continuity

WEEK 5 :
Completion of Metric Spaces, Compactness, Connectedness

WEEK 6 :
continuous functions

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,

Lebesgue covering lemma

WEEK 10 :
Products and compactness.

WEEK 11 :
metrizable spaces and topological spaces, heriditary properties

WEEK 12 :
Quotient topology, quotient spaces

WEEK 13 :
Connectedness, Hausdroff, Regular spaces & products

WEEK 14 :
Normal spaces, Lindelof spaces, Tietz Extension, Urysohn's lemma

WEEK 15 :

Tychonoff theorem, revision

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

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.