MATH30120 Introduction to Topology

Academic Year 2021/2022

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

Learning Outcomes:

On successful completion of this module, the student is expected to be able to understand the definitions, theorems and examples covered in all of the topics listed above. These include, but are not limited to, being able to define a topological space and reproduce examples like the discrete, indiscrete and cofinite topologies; give examples of topological spaces that do or do not satisfy the Hausdorff separation axiom; determine the set of accumulation points and the closure of a given subset; understand continuity and its many equivalent definitions; provide examples of continuous functions and homeomorphisms; construct products of finitely many topological spaces and understand continuity in this setting; construct common examples of quotient spaces; define compactness using open covers; show that closed bounded intervals of the real line are compact; show that products of finitely many compact spaces are compact; apply the Heine-Borel Theorem; define and compare connectedness and path connectedness using examples such as the topologist's sine curve; show that intervals of the real line are connected; show that products of finitely many connected spaces are connected; determine the connected components of a topological space; apply Brouwer's Fixed Point Theorem.

Student Effort Hours: 
Student Effort Type Hours
Lectures

30

Tutorial

10

Specified Learning Activities

24

Autonomous Student Learning

46

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
Continuous Assessment: Two class tests, both worth 10%. Throughout the Trimester n/a Standard conversion grade scale 40% No

20

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

80


Carry forward of passed components
No
 
Resit In Terminal Exam
Autumn Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing. 
Feedback Strategy/Strategies

• Group/class feedback, post-assessment

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