Topology

Module code: MA4144

When presented with a geometric object such as a ball, a wheel, a polyhedron (say a cube or dodecahedron), a knotted loop of string, or more exotic objects like Klein bottles and Möbius bands, there are several natural questions about them that one might ask. What is its length/area/volume? What symmetries does it have? How many sides are there? How many holes does it have? Some of these questions are quantitative: concerning properties which require measurements such as size or distance. On the other hand, some questions are qualitative: concerning properties which do not depend on the accurate shape of the object. For example, the Möbius band always has exactly one side no matter how much you stretch or deform it. Similarly, one cannot add or remove holes in a shape without cutting or gluing. Capturing and classifying qualitative properties which do not depend on explicit measurement is the essence of topology. The general theme is that of continuity, and topology asks what properties of a space are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. Such properties are known as topological invariants.

As a mathematical discipline (when compared with most other subjects on the undergraduate curriculum) topology is a modern field, originally developed in the late 19th and early 20th centuries. Although motivated by geometric objects, topological spaces are defined in terms of abstract sets, and can be very general. Through this abstraction, topological spaces have found a place in most branches of pure and applied mathematics. Topology one of the great unifying ideas of modern mathematics, providing a basic language for other areas, such as complex and functional analysis, measure theory and probability, algebra, and geometry. More recently, topology has become an important tool for studying and explaining many real-world phenomena. For example, it may be used to study instabilities in dynamical systems, equilibrium prices in economics, chemical and physical properties of molecules and particles, distributions of cosmological matter, and many other important problems of science. Topological methods also form a cornerstone for tackling modern problems (such as analysing large data sets) and in developing cutting-edge technologies (such as artificial intelligence and neural networks, robotics, liquid crystal displays, spintronic materials used in mass storage devices, and encoding information for quantum computers). 

In this module, we shall formalise the notion of a topological space, and investigate basic topological properties inherent to continuous deformations. A lot of the course is geometrical in flavour; however, we shall also make important links with other disciplines such as analysis and group theory. The content of this course forms a solid foundation in the subject of topology and will prepare you for more advanced topics and research active areas.