Aims & Learning Objectives:
Aims - This core course is intended to be an elementary and accessible introduction to the theory of metric spaces and the topology of 3n for students with both pure and applied interests.
Objectives - While the foundations will be laid for further studies in Analysis and Topology, topics useful in applied areas such as the Contraction Mapping Principle will also be covered. Students will know the fundamental results listed in the syllabus and have an instinct for their utility in analysis and numerical analysis.
Definition and examples of metric spaces. Convergence of sequences. Continuous maps and isometries. Sequential definition of continuity. Subspaces and product spaces. Complete metric spaces and the Contraction Mapping Principle. Picard's Theorem for u=f(t,u(t)). Sequential compactness, Bolzano-Weierstrass theorem and applications. Open and closed sets (with emphasis on 3n). Closure and interior of sets. Topological approach to continuity and compactness (with statement of Heine-Borel theorem). Equivalence of Compactness and sequential compactness in metric spaces. Connectedness and path-connectedness. Metric spaces of functions: C[0,1] is a complete metric space.