Topology and Metric Spaces - 2023 entry

Topology and metric spaces provide a set of powerful tools that are used in many other branches of mathematics (from Algebraic Topology and Numerical Analysis to Dynamical Systems and Ergodic Theory). Fundamental to these topics is the idea of generalising the idea of “closeness” of two objects in a set to a very general setting. These techniques are fundamental to the understanding of more advanced topics in mathematics such as Measure Theory, Functional Analysis, Algebraic Topology and Algebraic Geometry.

This course aims to give an introduction to topology and metric spaces as well as applications to basic concepts of measure theory. In every section covered in this course we will start by studying the definitions and then we will present examples and some basic properties. Some important theorems will be stated and proved. With this module you will have the opportunity to further refine your skills in problem-solving, axiomatic reasoning and the formulation of mathematical proofs.

Pre-requisite - MTH2001 or MTH2008

The objective of this module is to provide you an introduction to Topology and Metric Spaces. Our main objective will be to define the basic concepts clearly and to provide proofs of useful theorems.

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge

1. Recall and apply key definitions in Analysis;

2. State, prove and apply core theorems in Topology and metric spaces.

Discipline Specific Skills and Knowledge

3. Extract abstract problems from a diverse range of problems;

4. Use abstract reasoning to solve a range of problems.

Personal and Key Transferable / Employment Skills and Knowledge

5. Think analytically and use logical argument and deduction;

6. Communicate results in a clear, correct and coherent manner.

- Review of some real analysis: Real numbers, real sequences, limits of functions, continuity, intervals, set theory. (3 lectures)

- Metric spaces: Definition and examples, open and closed sets in metric spaces, equivalent metrics, examples. (4 lectures)

- Topological spaces: Bases, sub-bases and weak topologies, topologies of subspaces and products, homeomorphisms. (4 lectures)

- The Hausdorff condition: separation axioms, Hausdorff space, regular topological space. (3 lectures)

- Compact spaces: Definition, Compactness of [a,b], properties of compact spaces, continuous maps on compact spaces. An inverse function theorem. (3 lectures)

- Connected spaces: Connectedness, components, path-connectedness. (3 lectures)

- Complete metric spaces: Definition and examples, Fixed point theorems, the contraction mapping theorem. (4 lectures)

- Introduction to measure theory: Measure of plane sets. Outer and inner measure of a set. Measurable set (in the sense of Lebesgue). Some fundamental properties of Lebesgue measure and measurable sets. Definition and fundamental properties of measurable functions. (3 lectures)

- If time allows, a selection from the following: σ-algebras. Positive Borel measures. The Riesz representation theorem. Lp-spaces. Elementary Hilbert space theory. Banach spaces. Baire’s theorem. (3 lectures)
 
- Revision (3 lectures)

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 40%.

Basic reading:

ELE: http://vle.exeter.ac.uk/

Web based and Electronic Resources:

Other Resources:

Reading list for this module: