Content: To introduce the notions of Normed Space, Metric Space and Topological Space, and the fundamental properties of Compactness, Connectedness and Completeness that they may possess. Students will gain knowledge of definitions, theorems and calculations in:

• Normed, Metric and Topological spaces
• Open and closed sets and their relation to continuity
• Notions of Compactness and relations to continuous maps
• Notions of Connectedness and relations to continuous maps
• Notions of Completeness and relations to previous topics in the module

The module comprises the following chapters:

• Normed Spaces
• Metric Spaces
• Open and closed sets
• Continuity
• Topological spaces
• Compactness
• Connectedness
• Completeness

Learning Outcomes:

• Demonstrate understanding of the basic concepts, theorems and calculations of Normed, Metric and Topological Spaces
• Demonstrate understanding of the open-set definition of continuity and its relation to previous notions of continuity, and applications to open or closed sets
• Demonstrate understanding of the basic concepts, theorems and calculations of the concepts of Compactness, Connectedness and Completeness (CCC)
• Demonstrate understanding of the connections that arise between CCC, their relations under continuous maps, and simple applications

Books:

1. W A Sutherland, Introduction to Metric and Topological Spaces, OUP.
2. E T Copson, Metric Spaces, CUP.
3. W Rudin, Principles of Mathematical Analysis, McGraw Hill.
4. G W Simmons, Introduction to Topology and Modern Analysis, McGraw Hill. (More advanced, although it starts at the beginning; helpful for several third year and MMath modules in analysis).
5. A M Gleason, Fundamentals of Abstract Analysis, Jones and Bartlett.