Module code: MA3144
Topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. This can be studied by considering a collection of subsets of a given set called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include, among others, connectedness and compactness.
Topology can be formally defined as the study of qualitative properties of topological spaces that are invariant under a transformation given by a continuous map, especially those properties that are invariant under a certain kind of invertible transformation, called a homeomorphism. Homeomorphism can be considered the most basic topological equivalence of spaces. Intuitively, two spaces are homoeomorphic if one can be deformed into the other continuously without cutting or gluing.
Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of modern mathematics, and provides a basic language for other areas of mathematics, such as functional analysis, measure theory, algebraic topology, differential geometry and algebraic geometry. Recently topology has also become an important tool to study large data sets, genetic structures or neural networks. In this module we will study and investigate the basic properties of topological spaces and concepts inherent to them coming from continuous deformations.