Topology is a subject of fundamental importance in many branches of modern mathematics.  Basically, it concerns properties of objects which remain unchanged under continuous deformation, which means by squeezing, stretching and twisting, but not cutting.  Apples and oranges are topologically the same, but you can’t deform an orange into a doughnut! More precisely, we can never deform in a continuous way a sphere (the surface of an orange) into a torus (the surface of a doughnut). Knots are also examples of topological objects, where a trefoil knot can never be deformed into an unknotted piece of string. It's the business of topology to describe more precisely such phenomena.

The aim of this module is to explore properties of topological spaces. To distinguish topological spaces we will consider topological invariants such as the fundamental group, which is a powerful way of using an algebraic invariant to detect topological features of spaces.

Recommended Module: MA2013 Groups

• Understand the fundamental abstract notions of general topology including topological spaces, continuous maps, subspaces, connectedness, compactness, homeomorphisms, and separation properties.

• Understand homotopies of maps, homotopy equivalence and the construction of the fundamental group of a space. Understand basic properties of the fundamental group. Be able to compute the fundamental group of simple spaces.

• Understand applications of topological invariants to proving results in algebra and analysis

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Skills:
 

• Logical analysis and the ability to deal with new ideas and challenges with confidence

• Use of mathematical language in a logically and rhetorically correct way

• Understanding of the interactions between different mathematical disciplines (in this case Algebra and Topology)

• Application of abstract topological concepts in other areas of Mathematics

• Translation of graphical arguments into mathematical proofs

Transferable Skills:
 

• Assimilation and use of abstract ideas.

• Definition of a topological space, and examples.

• Continuous maps and homeomorphisms.

• Topological constructions: subspaces, products and quotients.

• Connectedness, Hausdorff spaces and compact spaces.

• Homotopy, and homotopy equivalence

• The fundamental group and some basic properties: moving the base point, products, functoriality, homotopy-invariance.

• Applications to: Brouwer fixed point theorem (for a disc), Borsuk-Ulam theorem, Ham sandwich theorem, fundamental theorem of algebra.

• Homology (if time permits)