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Week - 1 |
Definition and examples of metric spaces. Different metrics on R^n and the relationships between them. |
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Week - 2 |
Subspaces of metric spaces. |
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Week - 3 |
Lipschitz equivalent metrics. Metrics on function spaces. |
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Week - 4 |
Open balls, open sets, closed sets, bounded sets. |
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Week - 5 |
Continuous functions, pointwise and global continuity, isometries. |
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Week - 6 |
Interior, exterior, closure, and boundary of a set in metric spaces. |
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Week - 7 |
Sequences and convergence in metric spaces. |
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Week - 8 |
Cauchy sequences and complete spaces. |
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Week - 9 |
Pointwise convergence, uniform convergence. |
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Week - 10 |
Sequential compactness, uniform continuity, Bolzano–Weierstrass theorem. |
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Week - 11 |
Continuous functions on compact spaces, Extreme Value Theorem. |
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Week - 12 |
Contraction Mapping Theorem and some applications. |
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Week - 13 |
Topological spaces. |
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Week - 14 |
Metric topologies, topologically equivalent metrics. |