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Topology

University of Mumbai and UGC via Swayam

Overview



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 theory2. Analyze structure of Sets and other abstract algebraic structures3. Understand definitions; construct examples and counter examples based on definitions4. 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 functions2. Introduction to topology of metric spaces3. Introduction to topological spaces4. Continuity, convergence5. Subspaces, product spaces, quotient spaces6. Connectedness and Compactness

Syllabus

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

WEEK 8 :Product topology, separation axioms, Baire space

WEEK 9 :
Compactness

WEEK 10 :
Connectedness

WEEK 11 :
Quotient topology

WEEK 12 :
quotient spaces

WEEK 13 :
Lindel of spaces

WEEK 14 :
Urysohn's lemma

WEEK 15 :Tychonoff theorem, revision

Taught by

Dr. Mandar Bhanushe

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