General and Differential Topology in Graduate School level

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. --- (See attached file for full problem description)

General and Differential Topology in Graduate School level

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. --- (See attached file for full problem description)

Metrics

(See attached file for full problem description with equations) --- show that the function defines a metric on the Euclidean... ---

Metrics

(See attached file for full problem description with equations) --- Let X be a nonempty set and define the function by setting: a) show that the function d defines a metric on X, called the discrete metric. b) Determine the -balls for the discrete metric. c) What are the open subsets U of X with respect to the discrete metric?

Metrics

(See attached file for full problem description)