Course Syllabus

This syllabus outlines the modules of Metric Space.

📝Coursemeta: Course card

MJC-11 · 5 (50 Lectures + Tutorial Sessions) · 100 (70 External End-Semester Examination + 30 Continuous Internal Assessment)

📝Outcomes

CO1: Understand the basic concepts of metric spaces;
CO2: Correlate these concepts to their counter parts in real analysis;
CO3: Understand the abstractness of the concepts such as open balls, closed balls, compactness, connectedness etc. beyond their geometrical imaginations.

📝Module: Unit 1 — Metric Spaces, Open and Closed Sets | 12

Definition and examples of metric spaces, notion of open and closed ball, neighborhood of a point, open set, interior point, interior of a set, limit point of a set, derived set, closed set, closure of a set, diameter of a set, dense set, subspaces.

📝Module: Unit 2 — Completeness, Contraction Mapping, and Baire's Category Theorem | 12

Sequences in metric spaces, Cauchy sequences, complete metric space, Cantor's intersection theorem, Baire's category theorem, contraction mapping, Banach fixed point theorem.

📝Module: Unit 3 — Continuous Mappings, Uniform Continuity, and Homeomorphism | 10

Continuous mappings, sequential criterion and other characterizations of continuity, uniform continuity, homeomorphism.

📝Module: Unit 4 — Connectedness | 8

Connectedness, connected subsets of $\mathbb{R}$, connectedness and continuous mappings.

📝Module: Unit 5 — Compactness | 8

📝Textbooks

Kumaresan, S. (2014). Topology of Metric Spaces (2nd ed.). Narosa Publishing House. New Delhi.
Simmons, G. F. (2004). Introduction to Topology and Modern Analysis. Tata McGraw Hill. New Delhi.
Copson, E. T. (1968). Metric Spaces. Cambridge University Press.
Shirali, S. & Vasudeva, H. L. (2006). Metric Spaces. Springer.
Jain, P. K. & Ahmad, K. (2019). Metric Spaces. Narosa Publishing House. New Delhi.