| CODE | MAT5216 | ||||||
| TITLE | Extensions and Selections | ||||||
| UM LEVEL | 05 - Postgraduate Modular Diploma or Degree Course | ||||||
| MQF LEVEL | 7 | ||||||
| ECTS CREDITS | 15 | ||||||
| DEPARTMENT | Mathematics | ||||||
| DESCRIPTION | - Tietze's Extension Theorem (Continuous Extensions of Maps in Metric Spaces, Extension Formulas of Tietze, Poussin, Bohr, Hausdorff and Riesz); - Urysohn's Characterisation of Normality (Urysohn's Lemma, Urysohn's Families, Partitions of Unity, Absolute Retracts and Extensors); - Paracompactness and Continuous Extensions (Full Normality and Paracompactness, Dugundji's Extension Theorem, Arens' Extension Theorem); - Collectionwise Normality and Continuous Extensions (Collectionwise Normality, Normality and Absolute Retracts, Dowker's Extension Theorem); - Michael's Selection Theory (The Selection Problem, Selections and Extensions, Selections and Paracompactness, Bartle-Graves Theorem); - Collectionwise Normality and Continuous Selections (Extensions and Selections, Selections and PF-Normality, Selections and Hilbert Spaces). Study-unit Aims: In the mid 1950鈥檚 Ernest Michael wrote a series of fundamental papers relating familiar extension theorems to selections, thus laying down the foundation of the theory of continuous selections. Nowadays, selections became an indispensable tool for many mathematicians working in vastly different areas. The key importance of Michael鈥檚 selection theory is not only in providing a comprehensive solution to diverse selection problems, but also in the immediate inclusion of the obtained results into the general context of development of topology and analysis. This study-unit aims to to give a connected account of several extension and selection theorems that may lead young researchers comprehensively to up-to-date work in the area of continuous selections. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - Analyse basic techniques for constructing extensions of maps and selections; - Recognise the role of higher separation axioms in these constructions, and various applications in other areas; - Manage such fundamental concepts as normality, collectionwise normality, paracompactness, absolute extensors and retracts. 2. Skills: By the end of the study-unit the student will be able to: - Use ideas and methods for constructing extensions and selections; - Improve the handling of several topological techniques for constructing approximations of continuous maps such as covers and partitions of unity; - Be prepared to pursue advanced research problems in the area. Main Text/s and any supplementary readings: Lecture notes covering all topics. Textbooks: J. I. Nagata, Modern General Topology, Volume 33 (North-Holland Mathematical Library), Elsevier Science Publisher B.V., 1985. D. Repovs, P.V. Semenov, Continuous Selections of Multivalued Mappings, Volume 455 of Mathematics and Its Applications, Springer, 1998. |
||||||
| ADDITIONAL NOTES | Pre-requisite Qualifications: B.Sc. with Mathematics as a main area Follows from: MAT3219 and MAT3226 |
||||||
| STUDY-UNIT TYPE | Lecture and Independent Study | ||||||
| METHOD OF ASSESSMENT |
|
||||||
| LECTURER/S | Valentin Gutev |
||||||
|
The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2025/6. It may be subject to change in subsequent years. |
|||||||