Welcome to this Introduction to Topology!
Topology is the study of properties of spaces that are invariant under continuous deformations. An often cited example is that a cup is topologically equivalent to a torus, but not to a sphere. In general, topology is the rigorous development of ideas related to concepts such “nearness”, “neighbourhood”, and “convergence”.
The roots of topology go back to the work of Leibniz and Euler in the 17th and 18th century. It was only towards the end of the 19th century, through the work of Poincaré, that topology began taking shape as a subject in its own right. His seminal paper “Analysis Situs” from 1895 introduced, among other things, the idea of a homeomorphism and the fundamental group. Nowadays, topological ideas are an indispensable part of many fields of mathematics, ranging from number theory to partial differential equations.
In this course we will introduce topological spaces and study their properties. We will embark on a study of homotopy and introduce the fundamental group as an important tool to classify topological spaces. This page will feature weekly updates and pointers to additional material. Concise lecture notes will be published regularly, and weekly problem sheets will help you towards a better understanding.
See you all in the lectures, and enjoy!