Course layout
Week 1 : Chapter I - Introduction - Introduction, Normed linear spaces (NLS), Metric Spaces, ε -𝛿 Definition 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.
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