MATH315 - Analysis and Topology

Year

Credit points

Campus offering

Prerequisites

MATH104 Differential and Integral Calculus and one of the following: MATH203 Calculus of Several Variables , MATH205 Geometry , MATH220 Linear Algebra , MATH221 Applications of Mathematics , MATH224 Differential Equations and Mechanics

Incompatible

MATH306, MATH312 - Geometry and Topology

Unit rationale, description and aim

Calculus was developed to solve particular problems involving changing quantities, particular in mechanics. To begin with, many of the results and techniques used in calculus had no formal theoretical basis but were accepted purely because they seemed to work. Putting calculus on a firm foundation required several centuries to complete and provided a unified way to approach calculus as well as fundamental ideas that are applicable across mathematics. This unit will complete the study of calculus begun in MATH104 by providing proofs of many of the basic results.

In this unit students will be introduced to the foundations of calculus: sequences, series, limits of functions and continuous functions. Those basic ideas will be used to prove some basic properties of continuous functions and derivatives. Basic topology will be introduced. A construction of the real numbers will be described.

The aim of this unit is to provide students with an introduction to real analysis. That is, the ideas and results that support calculus. By meeting an outline of the definition of real numbers, students will also complete their study of the basic number systems.

Learning outcomes

To successfully complete this unit you will be able to demonstrate you have achieved the learning outcomes (LO) detailed in the below table.

Each outcome is informed by a number of graduate capabilities (GC) to ensure your work in this, and every unit, is part of a larger goal of graduating from ACU with the attributes of insight, empathy, imagination and impact.

Explore the graduate capabilities.

On successful completion of this unit, students should be able to:

LO1 - Prove simple results involving the limits of sequences and functions and summability of series (GA5, GA10)

LO2 - Prove that some simple functions are continuous (GA5, GA10)

LO3 - Solve simple theoretical problems using the more complicated properties of continuous functions (GA5, GA8)

LO4 - Use open sets to prove simple facts about continuous functions (GA5)

LO5 - Use least upper bounds to define the real numbers (GA5, GA8)

Graduate attributes

GA5 - demonstrate values, knowledge, skills and attitudes appropriate to the discipline and/or profession 

GA8 - locate, organise, analyse, synthesise and evaluate information 

GA10 - utilise information and communication and other relevant technologies effectively.

Content

Topics will include:

1. Sequences of real numbers, limits and convergence
2. Series, power series and summability
3. Functions and limits of functions (definition of limits)
4. Continuity
5. Continuous functions that change sign have zeros — Intermediate value theorem
6. Continuous functions are bounded on closed intervals and have maximum/minimum values
7. Differentiation, differentiability, mean value theorem
8. Open/closed (and other) sets in the real numbers
9. Properties of open sets, inverse images of sets, continuity via open sets
10. Least upper bounds and the real numbers

Learning and teaching strategy and rationale

As is common in Mathematics a variety of Active Learning approaches promote the best acquisition of skills and understanding. Lectures will typically be structured to provide explanations of the material to be covered, along with examples of the applications of that material, as well as formal and informal tasks for students that will reinforce that learning. Splitting the taught content of the lectures into short topics will ensure increased student engagement with the material, as well as providing opportunities for students to consolidate their learning before new material is covered. This allows students to learn the skills and then build understanding, competence and confidence via (ideally, face-to-face) tutorials involving cooperative groups, peer review and other relevant strategies. In all cases this should be supported using available online technology.

This unit will normally include the equivalent of 24 hours of lectures (typically 2 hours per week for 12 weeks) together with 24 hours attendance mode tutorials. Lectures will also be recorded and, where possible or required, students may have access to an online tutorial.

150 hours in total with a normal expectation of 48 hours of directed study and the total contact hours should not exceed 48 hours. The balance of the hours becoming private study.

Assessment strategy and rationale

To successfully complete an undergraduate Mathematics sequence, students need an understanding of a variety of basic Mathematical topics and an ability to apply that understanding to a variety of problems. To succeed at problem solving in Mathematics, students must have these skills at their fingertips and be able to recall them and choose an appropriate approach under some pressure. The assessment strategy chosen, while traditional, tests and supports student learning by providing opportunities to develop and test their problem-solving skills through the unit.

The continuous assessment component allows for the early detection of problems a student might be having and so ensures that appropriate guidance can be given early enough to support later learning in the unit. Students will be required to submit responses to questions dealing with simple problems in Analysis. As specified times, through the semester, students will submit responses to some of the questions to allow for the provision of feedback and learning support to students.

The examination components ensure that students have fully integrated the learning and can bring a variety of strategies to bear under pressure.

Typing of Mathematical notation either requires a significant investment of time, or knowledge of advanced Mathematical typesetting software. Neither of those skills are suitable for an undergraduate course in Mathematics. Consequently, assignments, tests and examinations are typically handwritten and either submitted as hardcopy or scanned and submitted to a dropbox, rather than through Turnitin. Students may choose to type their responses and submit them electronically, but this is not a requirement of the unit.

Overview of assessments

Representative texts and references

Brannan, D.A. (2018) A First Course in Mathematical Analysis Cambridge: Cambridge University Press

Flegg, H. (2012). From Geometry to Topology New York: Dover

Kelley, J., & Sloan, S. (2008). General Topology. Mountain View, CA: Ishi Press

Reid, M., Szendroi, B. (2005). Geometry and Topology Cambridge: Cambridge University Press

Singh, T.B. (2019) Introduction to Topology Singapore: Springer Verlag

Spivak, M. (2008) Calculus, 4th Edition, Publish or Perish

Stoll, M. (2021) Introduction to Real Analysis London: Taylor & Francis