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2012/13 Undergraduate Module Catalogue

MATH3224 Topology

15 creditsClass Size: 50

Module manager: Dr Peter Schuster
Email: pschust@maths.leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2012/13

Pre-requisite qualifications

MATH2015, or equivalent.

This module is mutually exclusive with

MATH3181

Inner Product and Metric Spaces

This module is approved as an Elective

Module summary

Topology, an invention of the twentieth century, is the study of the geometric and analytic properties of sets of points. Take an interval on the line. It may be open, closed, half-open, bounded, or unbounded. All these properties (except half-open) extend to more general sets (on the line or in other spaces) which it is the business of topology to investigate. How can we distinguish between sets that fall naturally into several pieces and those which do not? This module will tell. Another idea is that of distance in space. This generalises to distance between functions, which leads to efficient proofs of many, apparently unrelated, results.

Objectives

- To cover basic concepts of point set topology; to reinforce basic concepts of analysis; to provide a grounding for functional analysis, topology and differential geometry modules at levels 3 and 4;
- To improve students' powers of abstraction, problem-solving and visualisation.

On completion of this module, students should be able to:
(a) recall the basic definitions of point-set topology accurately
(b) write out proofs of the simpler theorems and propositions
(c) apply their knowledge to examples of specific topological and metric spaces

Syllabus

The topics covered are:
1. Abstract topological spaces - definition and examples, subspace topology.
2. Connectedness - connected components - characterisation of open sets in the real line.
3. Neighbourhoods, closed sets, closure, interior. Hausdorff spaces.
4. Continuous functions: definition and various criteria, homeomorphisms.
5. Continuity and connectedness; path-connectedness. The real line not homeomorphic to higher-dimensional spaces.
6. The Cantor space; space-filling curves.
7. Metric spaces. Equivalence. Sequences. Continuity in metric spaces. Normed spaces.
8. Complete metric spaces. Contraction Mapping theorem. Picard's Theorem. Implicit function theorem.
9. Compactness and sequential compactness. Characterisation of compact sets in Rn. Ascoli's theorem on compact sets in C(X). One-point compactification.
10. Characterization of compactness in terms of convergence.
11. Products of topological spaces. Tychonoff's theorem for finite and arbitrary products.

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	33	1.00	33.00
Private study hours			117.00
Total Contact hours			33.00
Total hours (100hr per 10 credits)			150.00

Private study

Studying and revising of course material.
Completing of assignments and assessments.

Opportunities for Formative Feedback

Regular problem solving assignments

Methods of assessment

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	2 hr 30 mins	100.00
Total percentage (Assessment Exams)		100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 08/01/2013

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