In these “Metric SpacesNotes PDF”, you will study the concepts of analysis which evidently rely on the notion of distance. In this course, the objective is to develop the usual idea of distance into an abstract form on any set of objects, maintaining its inherent characteristics, and the resulting consequences.
The topics we will cover will be taken from the following list:
Basic Concepts: Metric spaces: Definition and examples, Sequences in metric spaces, Cauchy sequences, Complete metric space.
Topology of Metric Spaces: Open and closed ball, Neighborhood, Open set, Interior of a set, Limit point of a set, Derived set, Closed set, Closure of a set, Diameter of a set, Cantor’s theorem, Subspaces, Dense set.
Continuity & Uniform Continuity in Metric Spaces: Continuous mappings, Sequential criterion and other characterizations of continuity, Uniform continuity, Homeomorphism, Contraction mapping, Banach fixed point theorem.
Connectedness and Compactness: Connectedness, Connected subsets of R, Connectedness and continuous mappings, Compactness, Compactness and boundedness, Continuous functions on compact spaces.
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