# MATH40350 Introduction to Descriptive Set Theory

Descriptive set theory is the study of certain classes of mathematical objects that can be defined or `described' explicitly. Lebesgue non-measurable subsets of real numbers and unbounded operators defined everywhere on Banach spaces are examples of objects that exist by virtue of the Axiom of Choice, but have no such explicit description.

Some key topics of classical descriptive set theory are covered, including

(1) metric spaces - review of fundamental concepts, Polish spaces and ways to construct them;
(2) trees - the tree A^{'less then N'} and the metric space A^N, the Baire and Cantor spaces, Lusin schemes, subtrees of A^{'less then N'} and retractions;
(3) Borel sets - sigma-algebras and measurable functions, ordinal numbers and the Borel hierarchy, universal sets;
(4) analytic and coanalytic sets - the separation theorem, complete analytic and coanalytic sets, e.g. IF, WF, DIFF, coanalytic ranks; (5) applications (time allowing).

The theory can be applied to many fields such as probability, functional analysis, potential theory, group representation theory, harmonic analysis and ergodic theory. It is seen, in an expanding area of research, as a natural framework for comparing the `complexity' or `difficulty' of various classification problems, in apparently very disparate fields of mathematics.

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Curricular information is subject to change

Learning Outcomes:

On successful completion of this module, the student is expected to demonstrate a clear understanding of the key topics listed above.

Student Effort Hours:
Student Effort Type Hours
Tutorial

12

Specified Learning Activities

60

Autonomous Student Learning

156

Total

228

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning.
Requirements, Exclusions and Recommendations

Not applicable to this module.

Module Requisites and Incompatibles
First courses on metric spaces and measure theory and integration (e.g. MATH30090 and MATH30360) are required. A first course on set theory (e.g. MATH40510) is recommended but not required.

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Examination: 50-minute written examination. Unspecified No Standard conversion grade scale 40% No

20

Examination: Written examination. 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

70

Continuous Assessment: Tutorial homework Throughout the Trimester n/a Standard conversion grade scale 40% No

10

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour