# Current students

Unit of study_

## Overview

Topology, developed at the end of the 19th Century to investigate the subtle interaction of analysis and geometry, is now one of the basic disciplines of mathematics. A working knowledge of the language and concepts of topology is essential in fields as diverse as algebraic number theory and non-linear analysis. This unit develops the basic ideas of topology using the example of metric spaces to illustrate and motivate the general theory. Topics covered include: Metric spaces, convergence, completeness and the Contraction Mapping Theorem; Metric topology, open and closed subsets; Topological spaces, subspaces, product spaces; Continuous mappings and homeomorphisms; Compactness Connectedness Hausdorff spaces and normal spaces. You will learn methods and techniques of proving basic theorems in point-set topology and apply them to other areas of mathematics including basic Hilbert space theory and abstract Fourier series. By doing this unit you will develop solid foundations in the more formal aspects of topology, including knowledge of abstract concepts and how to apply them. Applications include the use of the Contraction Mapping Theorem to solve integral and differential equations.

### Details

Academic unit Mathematics and Statistics Academic Operations MATH3961 Metric Spaces (Advanced) Semester 1, 2023 Normal day Remote 6

### Enrolment rules

 Prohibitions ? MATH4061 An average mark of 65 or above in 12cp from the following units (MATH2X21 or MATH2X22 or MATH2X23) None Real analysis and vector spaces. For example (MATH2922 or MATH2961) and (MATH2923 or MATH2962) Yes

### Teaching staff and contact details

Coordinator Daniel Daners, daniel.daners@sydney.edu.au

## Assessment

Type Description Weight Due Length
Supervised exam

Final exam
Written responses including mathematical arguments.
60% Formal exam period 2 hours
Outcomes assessed:
Assignment Assignment 1
Written work
12.5% Week 05
Due date: 24 Mar 2023 at 23:59

Closing date: 03 Apr 2023
Two weeks
Outcomes assessed:
multiple choice, short answer, some written respones
15% Week 08
Due date: 19 Apr 2023 at 23:59

Closing date: 19 Apr 2023
50 Minutes
Outcomes assessed:
Assignment Assignment 2
Written work
12.5% Week 11
Due date: 12 May 2023 at 23:59

Closing date: 22 May 2023
Two weeks
Outcomes assessed:
• Assignments:  There are two assignments. Each must be submitted electronically, as one single typeset or scanned PDF file only, via Canvas by the deadline. Note that your assignment will not be marked if it is illegible or if it is submitted sideways or upside down. It is your responsibility to check that your assignment has been submitted correctly and that it is complete (check that you can view each page). Late submisions will receive a penalty. A mark of zero will be awarded for all submissions more than 10 days past the original due date.
• Quiz: One quiz will be held online through Canvas. The quiz is 40 minutes and has to be submitted by the closing time of 23:59 on the due date. The quiz can be taken any time during the 24 hour period before the closing time.
• Final Exam: If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as a viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original exam. The format of the alternative assessment will be determined by the unit coordinator

### Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

Representing complete or close to complete mastery of the material.

Distinction

75 - 84

Representing excellence, but substantially less than complete mastery.

Credit

65 - 74

Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.

Pass

50 - 64

Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

Not meeting the learning outcomes to a satisfactory standard

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

### Special consideration

If you experience short-term circumstances beyond your control, such as illness, injury or misadventure or if you have essential commitments which impact your preparation or performance in an assessment, you may be eligible for special consideration or special arrangements.

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 Definition and basic examples of metric spaces including normed vector spaces. Limits and continuity, the topology of metric spaces. Lecture (3 hr)
Week 02 Topological notions: closed sets, interior, closure, boundary, derived set. Simple examples and common constructions of topologies. Topologically equivalent metrics. Lecture and tutorial (4 hr)
Week 03 Convergence of sequences, sequential characterisations of closed sets in metric spaces, local bases, first countable topological spaces, uniqueness of limits and introduction to separation axioms. Sequential characterisation of continuity. Lecture and tutorial (4 hr)
Week 04 Cauchy sequences and the completeness of metric spaces, uniform converbence Lecture and tutorial (4 hr)
Week 05 The contraction mapping theorem and applications: Existence and uniqueness of solutions to ordinary differential equations. Lecture and tutorial (4 hr)
Week 06 Uniform continuity, extension of uniformly continuous functions on dense subsets with applications, compact topological spaces Lecture and tutorial (4 hr)
Week 07 properties of continuous functions on compact sets, Lindelöf spaces and compactness in metric spaces Lecture and tutorial (4 hr)
Week 08 Characterisations of compact metric spaces and examples. Separable, second countable spaces Lecture and tutorial (4 hr)
Week 09 Initial and final topologies and applications, connected topological spaces Lecture and tutorial (4 hr)
Week 10 Connected components, continuous functions on connected sets, path connected sets Lecture and tutorial (4 hr)
Week 11 Normal spaces and the Urysohn lemma, Tieze Extension Theorem, Baire's theorem Lecture and tutorial (4 hr)
Week 12 Nearest point projections, orthogonal projections in Hilbert spaces. Lecture and tutorial (4 hr)
Week 13 Orthonormal systems and abstract Fourier series in Hilbert spaces, revision. Lecture and tutorial (4 hr)

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University’s graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

• LO1. Explain and apply the foundational ideas of point set topology
• LO2. Construct metric topologies, define open/closed sets of the metric spaces, and identify and explain the relationships between these and continuity and other properties of the metric space.
• LO3. Find and prove properties of several important classes of sequences in metric spaces
• LO4. Explain how knowledge from fundamental theorems of topological spaces and continuous mappings can be used to prove mathematical results
• LO5. Demonstrate a broad understanding of important concepts in topology and exercise critical thinking to identify and use concepts to analyse examples and draw conclusions
• LO6. Solve problems about differential equations using the contraction mapping theorem
• LO7. Apply the concepts of separable spaces and separation properties in both simple and complex examples.
• LO8. Write proofs and apply the theory of metric spaces to problems in topology.

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.

### Outcome map

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## Closing the loop

The unit was completely revised for 2022, and further adjustments will be made for 2023

### Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The University’s Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

General Laboratory Safety Rules

• No eating or drinking is allowed in any laboratory under any circumstances
• A laboratory coat and closed-toe shoes are mandatory