Unit of study_

# MATH4061: Metric Spaces

## Overview

At the end of this unit you will have received a broad introduction and gained a variety of tools to apply them within your further mathematical studies and/or in other disciplines.

### 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 Florica-Corina Cirstea, florica.cirstea@sydney.edu.au Florica-Corina Cirstea

## Assessment

Type Description Weight Due Length
Assignment Final take home exam
Take home exam
65% Formal exam period 16 pages
Outcomes assessed:
Assignment Assignment 1
Written work
10% Week 05 Two weeks
Outcomes assessed:
Assignment Quiz
Take home quiz
15% Week 09 16 pages
Outcomes assessed:
Assignment Assignment 2
Written work
10% Week 11 Two weeks
Outcomes assessed:

### Assessment summary

• one take home quiz to provide written answers to short questions
• two assignments to provide written solutions to questions. Detailed information for each assessment can be found on Canvas.

Detailed information for each assessment can be found on Canvas.

### 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

When you don’t meet the learning outcomes of the unit 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.

This unit has an exception to the standard University policy or supplementary information has been provided by the unit coordinator. This information is displayed below:

Late submission of assignments will not be considered. The mark for each assignment will be recorded only if the assignment has been submitted before the deadline. The better mark principle will be applied for each assignment.

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
Multiple weeks Topological and metric concepts Lecture and tutorial (12 hr)
Fundamental results on topological spaces, continuous mappings and their properties Lecture and tutorial (12 hr)
Topological concepts: compactness and connectedness Lecture and tutorial (12 hr)
Basic Hilbert space theory and applications Lecture and tutorial (12 hr)
Week 01 Introduction to metric spaces Lecture (3 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