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Swayam

Topology

University of Mumbai and UGC via Swayam

Overview

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 theory2. Analyse 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 logicWe shall cover the following topics in this one-semester 3 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

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.

Taught by

Dr. Mandar Bhanushe

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