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School of Mathematics

MATH20122 Online Course Materials

IMPORTANT NOTICE

The usual three classes will take place on Tuesday 30 April 2013, and will focus on revision. THE TWO CLASSES ON FRIDAY 3 MAY (which I announced last week) WILL NOT TAKE PLACE, because I have now realised that I will be away.

By way of recompence, here are the solutions to the 2012 exam. PLEASE NOTE THAT the syllabus changed in January 2011 when I took over the course, so there will be topics on earlier papers (2010 and before) that are not now required.

Feel free to email me if you need additional help.

Introduction

Please read this introduction to the lecture notes before the course begins.

Here is the course timetable.

Lecture Notes and Problems

Each chapter of the notes includs the problems for discussion in the weekly feedback classes.

Solutions to Problems

The solutions will only be posted after sufficient time has elapsed for serious contemplation of each week's problems.

Corrections

• If corrections to the notes are required, they will be posted here.

Week 4 Test

Here is the 2013 test and solutions, together with my feedback.

The 2013 Examination Paper

The structure of the 2013 examination paper will be similar to those of 2010, 2011 and 2012, with the same number of questions and corresponding allocations of marks.

The structure of the August resit will be very close to that of the May original.

The subject matter of this course is main-stream pure mathematics, and cannot exist (nor be applied) without rigorous proofs. The examination paper will therefore ask for a small number of proofs from the lecture notes, although they will only be the type which input definitions and certain data, and rearrange them to produce the required output. As an aid to understanding this requirement, the following list includes proofs that will be required for one (or both!) of the May and August examinations 2013.

• Every metric given in lectures or problems satisfies all three axioms
• Any open ball is contained in the closed ball with the same centre and radius; and any closed ball is contained in or equal to an open ball with the same center and larger radius
• Any open ball is open and any closed ball is closed, in any metric space
• Properties of interiors, open sets, closures, and closed sets implied by the definitions
• Every isometry is continuous
• Any continuous function between metric spaces maps convergent sequences to convergent sequences, and preserves their limits
• A function is continuous iff the inverse image of every open set is open
• A discrete metric space is compact iff it is finite, and any closed subset of a compact metric space is compact
• The Cantor set K is compact; and a given number lies (or does not lie) in K
• A contraction is continuous; and any contraction on a complete metric space has a unique fixed point.
• Any contraction on a complete metric space has a unique fixed point.

Some of the proofs that are hidden amongst the Problems may also be requested; for example, Problem 42 requires a proof that π/4 does not lie in the Cantor set. Similar warnings apply to statements that are labelled as Examples in the lecture notes.

Proofs that will not be required under any circumstances are:

• Any closed interval of the Euclidean line is compact
• If a subset of n-dimensional Euclidean space is closed and bounded, then it is compact.