Unit of study_

# MATH4061: Metric Spaces

## 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.

### Unit details and rules

Unit code MATH4061 Mathematics and Statistics Academic Operations 6 MATH3961 An average mark of 65 or above in 12cp from the following units (MATH2X21 or MATH2X22 or MATH2X23 or MATH3061 or MATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3962 or MATH3963 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979) None Real analysis and vector spaces. For example (MATH2922 or MATH2961) and (MATH2923 or MATH2962) Yes

### Teaching staff

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

## Assessment

Type Description Weight Due Length
Final exam (Take-home short release) Final take-home exam
Written responses including mathematical arguments.
60% Formal exam period 2 hours
Outcomes assessed:
Assignment Assignment 2
Written work
12.5% Please select a valid week from the list below
Due date: 13 May 2022 at 23:59

Closing date: 23 May 2022
Two weeks
Outcomes assessed:
Assignment Assignment 1
Written work
12.5% Week 05
Due date: 25 Mar 2022 at 23:59

Closing date: 04 Apr 2022
Two weeks
Outcomes assessed:
Written responses including mathematical arguments.
15% Week 08
Due date: 13 Apr 2022 at 23:59

Closing date: 13 Apr 2022
50 Minutes
Outcomes assessed:
= Type D final exam

### Assessment summary

• 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. Further extensions past this time will not be permitted.
• 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.

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.

## Learning support

### Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

### Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

### Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 Definition and basic examples of metric spaces including normed vector space. 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, the completion of a metric space. Topological versus metric properties. Lecture and tutorial (4 hr)
Week 05 Compact topological spaces, properties of continuous functions on compact sets. Characterisations of compact metric spaces. Lecture and tutorial (4 hr)
Week 06 More on compactness. Separable, second countable and Lindelöf spaces. Lecture and tutorial (4 hr)
Week 07 Uniform continuity, extension of uniformly continuous functions on dense subsets. Lecture and tutorial (4 hr)
Week 08 The contraction mapping theorem and applications: Existence and uniqueness of solutions to ordinary differential equations, inverse and implicit function theorems. Lecture and tutorial (4 hr)
Week 09 Connected topological spaces, connected components, continuous functions on connected sets. Lecture and tutorial (4 hr)
Week 10 Normal spaces, Urysohn's Lemma and the Tieze Extension Theorem. Lecture and tutorial (4 hr)
Week 11 Baire's theorem and applications Lecture and tutorial (4 hr)
Week 12 Hilbert spaces, orthogonal projections and abstract Fourier series. Lecture and tutorial (4 hr)
Week 13 More on Hilbert space theory, 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 rigorously prove properties of several important classes of sequences in metric spaces
• LO4. Synthesise knowledge from fundamental theorems of topological spaces and continuous mappings and use this to prove new results
• LO5. Demonstrate a broad and deep understanding of important concepts in topology and exercise critical thinking to identify and use these concepts to analyse examples and create novel 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 transfer the theory of metric spaces to other areas of mathematics.

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

## Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

No changes have been made since this unit was last offered.

### 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