| Course Archives Theoretical Statistics and Mathematics Unit | ||||||||
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Course: Topology I Instructor: Aniruddha C Naolekar Room: G25 Level: Postgraduate Time: Currently offered |
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| Syllabus Past Exams Syllabus: i) Topological spaces, open and closed sets, basis, closure, interior and boundary. Subspace topology, Hausdorff spaces. Continuous maps: properties and constructions; Pasting Lemma. Homeomorphisms. Product topology. ii) Connected, path-connected and locally connected spaces. Lindel of and Compact spaces, Locally compact spaces, one-point compactification and Tychonoffs theorem. Paracompactness and Partitions of unity (if time permits). iii) Countability and separation axioms., Urysohn embedding lemma and metrization theorem for second countable spaces. Urysohns lemma, Tietze extension theorem and applications. Complete metric spaces. Baire Category Theorem and applications. iv) Quotient topology: Quotient of a space by a subspace. Group action, Orbit spaces under a group action. Examples of Topological Manifolds. v) Topological groups. Examples from subgroups of GLn(R) and GLn(C). vi) Homotopy of maps. Homotopy of paths. Fundamental Group. Suggested Texts : (a) J. R. Munkres, Topology: a first course, Prentice-Hall (1975). (b) G.F. Simmons, Introduction to Topology and Modern Analysis, TataMcGraw- Hill (1963). (c) M.A. Armstrong, Basic Topology, Springer. (d) J.L. Kelley, General Topology, Springer-Verlag (1975). (e) J. Dugundji, Topology, UBS (1999). (f) I. M. Singer and J. A. Thorpe, Lecture notes on elementary topology and geometry, UTM, Springer. Evaluation:
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