Topology concerns properties of objects which remain unchanged under continuous deformation, which means by squeezing, stretching, and twisting, but not cutting or gluing. For this reason, it is sometimes called "rubber-sheet geometry". Knots are also examples of topological objects, where a trefoil knot can never be deformed into an unknotted piece of string. Algebraic topology gives you a rigorous way to prove that we can never deform in a continuous way a sphere (the surface of a ball) into a torus (the surface of a doughnut). It has become a subject of fundamental importance in many branches of modern mathematics including mathematical physics and data science.

This module introduces you to the basic concepts of point-set topology and algebraic topology including topological spaces and their invariants as well as the fundamental group. You will know by the end of the course how to determine if a topological space has basic topological properties such as connectedness and compactness as well as how to compute fundamental groups.

Prequisite modules: MA1005, MA1006, 

Recommended Module: MA2014 Groups

• Understand the fundamental abstract notions of point-set topology including topological spaces, continuous maps, subspaces, connectedness, compactness and homeomorphisms.
• 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.

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 to support the learning (e.g. videos, exercise sheets, lecture notes, quizzes)

Students are also expected to undertake at least 50 hours of self-guided study throughout the duration of the module, including preparation of formative assessments.

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.

• Application of abstract topological concepts in other areas of Mathematics

• Translation of graphical arguments into mathematical proofs

• Definition of a topological space, and examples.
• Continuous maps and homeomorphisms.
• Topological constructions: subspaces, products and quotients.
• Compactness and Hausdorff spaces.
• Connectedness and path-connectedness.
• Homotopies and homotopy equivalence
• The fundamental group and some basic properties: moving the base point, products, functoriality, homotopy-invariance.
• Applications