The course will start with definition of metric spaces and topological spaces and proceeds to study topological aspects of metric spaces. W e prove three very important theorems on complete metric spaces and give construction of completion of metric spaces. The course then takes up the study of topological spaces, constructing new topologies from the old. Bases and subbases, I and II countability, separability, connected and path connectedness, Compactness , Lindeloffness, separation axioms etc. The course ends with the celebrated results such as Urysohn’s lemma and Titze’s extension theorem and some applications. The content of this course is mandatory for any meaningful study of analysis and further topology. The lecture slides are backed up by full notes, a strong team of tutors who will handle all the queries sympathetically and also by a number of online interactive session. PRE-REQUISITE : Anybody who has passed 12 standard can take this course though one course in real analysis available on NPTEL Portal is preferable. INDUSTRY SUPPORT : Nil. INTENDED AUDIENCE : Anybody who has passed 12th standard and had one or two courses in Real Analysis. Students from various streams which include some modern mathematics such as B. Sc., M. Sc., Students, Ph. D. , B. Tech. and M. Tech. who had not attended any topology courses seriously, before this, will be able to benefit form this course.

Syllabus

Week 1 :Chapter I - Introduction -Introduction,Normed linear spaces (NLS),Metric Spaces,ε -𝛿Definition of continuity,Examples of continuous functions,Topological Spaces.
Week 2 :Chapter I - Introduction -Examples,Functions,Topology of the n-dim. Euclidean space,Equivalences on metric spaces,Equivalences continued.
Week 3 :Chapter I - Introduction -Counter examples,Definitions and examples,Closed sets,Interiors and boundaries,Interiors and derived sets.
Week 4 :Chapter I - Introduction -More examples, Metric Trinity, Baire’s Category Theorem, An Application in Analysis, Completion of Metric space.
Week 5 :Chapter II - Creating New Spaces -Bases and subbases,Subbases,Box Topology,Subspaces,Union of spaces.
Week 6 :Chapter II - Creating New Spaces -Extending neighbourhoods, Quotient Spaces, Product of spaces, Study of Products - continued, Induced and co-induced topologies.
Week 7 :Chapter III- Smallness Properties of Topological Spaces -Path Connectivity, Connectivity, Connected components, Connectedness-continued, Local Connectivitym, More Examples.
Week 8 :Chapter III- Smallness Properties of Topological Spaces -Compactness and Lindelöfness, Compact Metric Spaces, Compactness-continued, Countability and Separability, Types of Topological Properties.
Week 9 :Chapter III- Smallness Properties of Topological Spaces -Productive Properties, Productive Properties-continued, Tychonoff Theorem, Proof Alexander’s Subbase Theorem.
Week 10 :Chapter IV - Largeness properties -Fréchet Spaces, Hausdorff spaces, Examples and Applications, Examples and Applications - continued.
Week 11 :Chapter IV - Largeness properties -Regularity and Normality, Characterization of Normality, Tietze’s Characterization of Normal Spaces, Productiveness of Separation Axioms, The Hierarchy.
Week 12 :Chapter V - Topological groups and Topological Vector Spaces -Topological Groups, Topological Groups-continued,Topological Groups-continued,Topological Vector Spaces,Topological Vector Spaces-continued,Topological Vector Spaces-continued.