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# General and Differential Topology

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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/general-and-differential-topology-49366

#### Solution Preview

(a) If A is a basis of topology T on X, than

(1) The intersection of all topologies that contain A cannot be larger than T because T is included in the "all".
(2) It cannot be smaller ...

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