# 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).

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Curricular information is subject to change

Not recorded
Student Effort Hours:
Student Effort Type Hours
Specified Learning Activities

24

Autonomous Student Learning

46

Lectures

30

Tutorial

10

Total

110

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning.
Requirements, Exclusions and Recommendations
Learning Requirements:

It is necessary for students to have taken MATH30090 in order to take this module.

Module Requisites and Incompatibles
Pre-requisite:
MATH10320 - Mathematical Analysis

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade In Module Component Repeat Offered
Class Test: Two class tests, both worth 10%. Throughout the Trimester n/a Standard conversion grade scale 40% No

20

No
Examination: 2-hour written exam 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

80

No

Carry forward of passed components
No

Remediation Type Remediation Timing
In-Module Resit Prior to relevant Programme Exam Board