# MATHS 350 : Topology

## Science

### Course Prescription

Aspects of point-set, set-theoretic and algebraic topology including: properties and construction of topological spaces, continuous functions, axioms of separation, countability, connectivity and compactness, metrisation, covering spaces, the fundamental group and homology theory. Recommended preparation: MATHS 333.

### Course Overview

This course introduces important concepts for postgraduate study in geometry and analysis, and provides a good background for an honours or postgraduate project in topology. Students taking this course will obtain first-hand experience in research collaboration by working in groups to prove theorems and construct counterexamples. These skills will be invaluable for postgraduate study in pure mathematics and for any career involving mathematics research. Maths 350 covers the same topics as Maths 750, but at a level suited for level 3 undergraduate study. This is reflected in assignment and exam questions. Maths 350 are expected to attend weekly tutorials and will sit a mid-semester test.

### Course Requirements

Prerequisite: MATHS 332 and Departmental approval Restriction: MATHS 750

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity Capability 6: Social and Environmental Responsibilities

### Learning Outcomes

By the end of this course, students will be able to:
1. An ability to present precise definitions of a range of important topological concepts (Capability 1)
2. An ability to describe important theorems connecting these topological properties (Capability 1 and 2)
3. An ability to present proofs of important topological theorems (Capability 1)
4. An ability to determine the truth or otherwise of statements involving the topological properties referred to above, and to provide proof (Capability 1, 2, 3 and 5)
5. An ability to construct examples illustrating that certain statements involving topological properties are false (Capability 1, 2, 3 and 5)
6. An ability to communicate topological concepts, theory and proofs (Capability 1, 2, 3, 4 and 5)
7. An ability to evaluate and appreciate the work of peers (Capability 1, 2, 3, 4, 5 and 6)
8. An ability to collaborate with others (Capability 1, 2, 3, 4, 5 and 6)
9. An ability to apply topological ideas to the proof of applications such as the Intermediate Value Theorem, Brouwer’s Fixed Point Theorem, the non-existence of vector fields on spheres of even dimension (Capability 1, 2, 3, 4 and 5)

### Assessments

Assessment Type Percentage Classification
Assignments 15% Individual Coursework
Reports 10% Individual Coursework
Course engagement 5% Group Coursework
Test 20% Individual Test
Final Exam 50% Individual Examination
1 2 3 4 5 6 7 8 9
Assignments
Reports
Course engagement
Test
Final Exam

### Learning Resources

Course notes made available on Canvas

### Special Requirements

N/A

This course is a standard 15 point course and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in.

For this course, you can expect 36 hours of lectures/course engagement,  10 hours of tutorials, 24 hours of reading and thinking about the content and 24 hours of work on assignments and/or test preparation.

### Digital Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

### Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website at http://disability.auckland.ac.nz

### Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page: https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.

This should be done as soon as possible and no later than seven days after the affected test or exam date.

### Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

### Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter (https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html).

### Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.

Published on 08/07/2020 12:26 p.m.