MATH30120 Topology

Academic Year 2023/2024

Topology is, primarily, the study of continuity and the properties of a space that remain unchanged after the action of continuous functions. The idea of a continuous function is central to the study of calculus and mathematical analysis. As functions came to be defined on spaces much more complicated than the real line, the plane, 3-space and so on, it became necessary to formulate a definition of continuity that could be applied in as many situations as possible. Following the revolutionary ideas of Cantor in the 1880s and the invention of set theory, progress in this regard came swiftly with the introduction of 'metric spaces' in 1906, the more general 'topological spaces' in 1914, and the corresponding definitions of continuity, by Fréchet and Hausdorff, respectively.

Topology is fundamental to many of the concepts in modern mathematics and physics, and this module should provide students with the essential tools necessary to understand and investigate these concepts. We will cover topics listed below. Time allowing, we will cover Brouwer's Fixed Point Theorem, which was used by John Nash in his seminal work on game theory, for which he was awarded the Nobel Prize in Economics.

1. topological spaces;
2. bases and subbases, separation axioms;
3. closed sets, accumulation points, closures;
4. continuous functions, homeomorphisms;
5. product and quotient spaces;
6. compact spaces;
7. connected and path connected spaces, connected components, and
8. Brouwer's Fixed Point Theorem (time allowing).