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MA260 Norms, Metrics and Topologies

Lecturer: Andras Mathe

Term(s): Term 2

Status for Mathematics students: Core

Commitment: 30 lectures

Assessment: 100% 2-hour June examination

Formal registration prerequisites: None

Assumed knowledge:

MA131 Analysis I & II or MA137 Mathematical Analysis I & II:

  • Sequences
  • Convergence
  • Cauchy sequences
  • Series
  • Continuous functions
  • Differentiable functions

MA132 Foundations or MA138 Sets and Numbers:

  • Set theory
  • Proofs
  • Cardinality

MA244 Analysis III or MA258 Mathematical Analysis III:

  • Pointwise and uniform convergence of sequences of functions

Useful background:

MA259 Multivariable Calculus:

  • Open and closed sets in R^n

Synergies: The following module goes well together with Norms, Metrics & Topologies:

MA243 Geometry

Leads to: The following modules have this module listed as assumed knowledge or useful background:

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. B. Mendelson (1991) Introduction to Topology. Dover.

Additional Resources