Mathematics

Topology and Advanced Analysis (L6)

Module code: G1026
Level 6
15 credits in autumn semester
Teaching method: Lecture
Assessment modes: Coursework, Unseen examination

Topology has a central role in all of mathematics and having a proper understanding of its basic ideas, concepts and main results is essential as part of a good mathematics degree.

You will be introduced to basic concepts and properties of Topological Spaces. Various applications to Functional Analysis, Real and Complex Analysis will be discussed and there will be a fair balance between theory and examples.

Topics include separation axioms, metrisability, compactness and coverings, total boundedness, nets and Lebesgue numbers, Arzela-Ascoli theorem and compactness in function spaces, Montel's theorem and normal families, Tychonoff's theorem and infinite products and upon time allowing the Banach-Alaoglu theorem and compactness in weak-star topologies.

Module learning outcomes

• Gain a coherent and detailed knowledge of the concepts of a topological space and open and closed sets and interior and closure.
• Systematically understand the concept of a metric and a metrisable topology.
• Be able to use and have a coherent knowledge of the concepts of coverings, continuity and compactness.
• Systematically understand Arzelà-Ascoli theorem and some of its basic consequences.