Basic Concepts: Definition and Examples of Metric Spaces, Distance between Sets and Diameter of a Set, Open Sets and Interior Points, Closed Sets and Closure of Sets, Subspaces.

Complete Metric Spaces and Separable Spaces: Convergent Sequences, Cauchy Sequences, Complete Metric Spaces, Completion. Separable Spaces, Countability, Dense Sets, Nowhere Dense Sets.
Definitions and Basic Concepts, Sequentially Compact Spaces,Totally Bounded Spaces.

Continuous Functions: Definition and Characterizations, Continuous Functions and Compact Spaces, Uniform Continuous Functions, Homeomorphism and Equivalent Metrics, Uniform Convergence of Sequences of Functions.

Connected Spaces: Separated Sets, Connected Sets, Continuous Functions and Connected Sets, Components.

1. Qamrul Hasan Ansari, “Metric Spaces: Including Fixed Point Theory and Set-valued Maps” Alpha Science International, Ltd, 2010.
2. Mícheál Ó Searcóid, “Metric Spaces”; Undergraduate Texts in Mathematics, Springer Verlag London Limited, 2007.
3. E. T. Copson, “Metric spaces”, Cambridge Tracts in Mathematics, Cambridge University Press, 1988.
4. Tom M. Apostol Mathematical Analysis Narosa publishing house, New Delhi, 2nd Ed. 1989.