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    Countable collections

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    This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics.

    Content: Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differentiable manifolds, tangent spaces, vector fields, submanifolds, inverse function theorem, immersions, submersions, partitions of unity, Sard's theorem, embedding theorems, transversality, classification of surfaces.

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    https://brainmass.com/math/geometry-and-topology/countable-collections-49368

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    There are two proofs in this solution regarding countable collections.

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