## MATH3224 Topology

### 15 creditsClass Size: 100

Module manager: Dr J.M. Speight
Email: speight@maths.leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

### 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.
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.
8. Completion of a metric space.
9. Contraction Mapping theorem. Picard's Theorem. Implicit function theorem.
10. Hausdorff spaces and normal spaces.
11. Compactness and sequential compactness. Characterisation of compact sets in Rn. Ascoli's theorem on compact sets in C(X). One-point compactification.
12. Nets (or filters) in a topological space. Continuity properties in terms of nets. Weak topologies. Examples to show sequences are not always adequate.
13. Characterization of compactness in terms of convergence.
14. Products of topological spaces. Tychonoff's theorem for finite and arbitrary products.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Example Class 7 1.00 7.00 Lecture 26 1.00 26.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 100.00 Total percentage (Assessment Exams) 100.00

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