Syllabus
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Course Code: MMATH21-304 Course Name: Elective - I ) Advanced Topology |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Convergence of sequences in topological spaces and in first axiom topological spaces, Nets in topological spaces, convergence of nets, Hausdorffness and convergence of nets, Subnets and cluster points, canonical way of converting nets to filters and vice versa, their convergence relations ( Scope as in theorems 2-3,5-8 of Chapter 2 of Kelley’s book recommended at Sr. No.1) Connected spaces, connected subspaces of the real line, components and local connectedness (Scope as in relevant portions of sections 23-26 of Chapter 3 of the book by ‘Munkres’ recommended at Sr. No. 2) |
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| 2 | Definition and examples of metrisable spaces, Urysohn’s metrisation theorem. Locally finite family, its equivalent forms, countably locally finite family, refinement, open refinement, closed refinement of a family, existence of countably locally finite open covering of a metrisable space, Nagata-Smirnov metrisation theorem, Paracompactness, normality of a paracompact Hausdorff space, paracompactness of a metrisable space and of regular Lindelof space, Smirnov metrisation theorem. (Scope as in theorems 34.1, 39.1-39.2. 40.3, 41.1-41.5 and 42.1 of Chapter 6 of the book by ‘Munkres’ recommended at Sr. No. 2) |
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| 3 | Relation of homotopy of paths based at a point and homotopy classes, product of homotopy classes, Fundamental group, change of base point topological invariance of fundamental group. (scope as in relevant parts of Chapter IV of the book by ‘Wallace’ recommended at Sr. No.3) Euclidean simplex, its convexity and its relation with its faces, standard Euclidean simplex, linear mapping between Euclidean simplexes of same dimension (scope as in relevant parts of Chapter V of the book by ‘Wallace’ recommended at Sr. No.3) |
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| 4 | Singular simplexes and group of p-chains on a space, special singular simplex on and its boundary, induced homomorphism between groups of chains, boundary of a singular simplex and a chain, cycles and boundaries on a space, homologous cycles, homology and relative homology groups, induced homomorphism on relative homology groups, induced homomorphism on relative homology groups, topological invariance of relative homology groups, Prisms, homotopic maps and homology groups. (scope as in relevant parts of Chapter VI of the book by ‘Wallace’ recommended at Sr. No.3) Join of a point and a chain, Barycentric subdivision operator B, diameter of a Euclidean simplex and a singular simplex, operator H and its relation with B, representation of an element of a relative cycle made up of singular simplexes into members of a given open cover of the space, the excision theorem (scope as in relevant parts of Chapter VII of the book by ‘Wallace’ recommended at Sr. No.3) |
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