Geometry and Topology

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Understand the Contraction Mapping Theorem and use it to solve a variety of fixed point problems including proving the existence of solutions of differential equations.
• Understand the behaviour and properties of sequences in general metric spaces, including convergence, and the Cauchy property.
• Understand the main properties of subsets of topological spaces including criteria for the set to be open, closed, compact, connected, path connected
• Be familiar with the notion of continuity of functions between metric spaces condition, and general topological spaces, including isometries, Lipschitz maps, homeomorphisms and quotient maps.
• Understand properties of metrics on a wide range of spaces and be able to check if a given real valued function of two variables is a metric on a set.

Metrics on a set:

• Definitions and main examples

oMetrics on Rn

oNormed vector spaces as metric spaces

oSpaces of sequences

oSpaces of continuous functions

oLength spaces

Geometry:

• Isometries
• Euclidean geometry and classification of Euclidean isometries
• Further examples of geometries

Properties of metric spaces:

• Sequences and convergence, completeness
• The Bolzano-Weierstrass theorem
• Continuous functions of metric spaces
• Lipschitz functions and contractions
• Uniform and pointwise convergence in spaces of continuous functions of real variable
• The Contraction Mapping Theorem
• Picard’s Theorem and differential equations

Topology:

• Definition of topology, basis of topology
• Metric topology, open and closed sets in metric spaces
• Interior and closure
• Continuous functions and homeomorphisms
• Connected and path connected sets
• Compact sets
• Tychonoff’s theorem and its consequences, including the Heine-Borel theorem
• Outline of quotient topology and basic quotient constructions.

Lectures, problem classes, workshops, private study

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.