Metric Spaces

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Metric Spaces

Course Objectives: The course aims at providing the basic knowledge pertaining to metric spaces such as open and closed balls, neighborhood, interior, closure, subspace, continuity, compactness, connectedness etc.

Course Learning Outcomes: The course will enable the students to:

1. Understand the basic concepts of metric spaces;

2. understand the abstractness of the concepts such as open balls, closed balls, open sets, closed sets, derived sets and their geometrical imaginations.

3. understand the concepts of boundedness, compactness, and connectedness.

Course Contents:

Basic Concepts of Metric spaces: Definition of metric space and its examples, Sequences in metric spaces, limit of a sequence, Cauchy sequences, Complete metric space and its examples.

Topology of Metric Spaces: Open and closed ball, Neighborhood, Open sets, interior point, 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 in metric spaces and related examples.

Connectedness and Compactness: Connectedness, Connected subsets of R, Connectedness and continuous mappings, Compactness and boundedness.

Reference:

1. Simmons, G. F. (2004). Introduction to Topology and Modern Analysis. Tata McGraw Hill. New Delhi.

2. Rudin, W. (1976). Principle of Mathematical Analysis. Tata McGraw Hill.

3. Jain, P.K and Ahmed K. (1996). Metric Spaces, Narosa Publ. House, New Delhi.

4. Shirali, Satish & Vasudeva, H. L. (2009). Metric Spaces, Springer, First Indian Print.

5. Kumaresan, S. (2014). Topology of Metric Spaces (2nd ed.). Narosa Publishing House. New Delhi.

6. Copson, E.T. (1996). Metric Spaces, Universal Book Stall, New Delhi.