| CODE | MAT3220 | ||||||
| TITLE | Introduction to Dimension Theory | ||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||
| MQF LEVEL | Not Applicable | ||||||
| ECTS CREDITS | 5 | ||||||
| DEPARTMENT | Mathematics | ||||||
| DESCRIPTION | (I) Elementary Combinatorial Techniques: • Affine Notions (linear, affine and convex combinations, linear and affine independence) • Simplexes (n-dimensional simplex, affine coordinates, vertices and faces, geometric simplex, boundary and barycenter) • Triangulation (simplicial complex, geometric realization, triangulation, barycentric triangulation) • Simplexes in Euclidean Spaces (faces of n-dimensional simplexes, Sperner's lemma) • Applications (Brouwer Fixed-Point Theorem, Non-Retraction Theorem, Non-Homogeneity Theorem, Knaster–Kuratowski–Mazurkiewicz Theorem) (II) Elementary Dimension Theory: • The Covering Dimension (essential and inessential families of pairs of sets, dimension of Euclidean spaces, partitions and dimension of partitions) • Zero-Dimensional Spaces (products of 0-dimensional spaces, 0-dimensional subspaces of the real line, Cantor middle-third set, totally disconnected spaces) • Open Covers (swelling and shrinking, order of a cover, dimension and covers) • Basic Theorems (The Countable Closed Sum Theorem, The Subspace Theorem, The Classification Theorem) • The Imbedding Theorem (Nobeling's universal n-space, applications) • Inductive Dimension Functions (ind and Ind, The Coincidence Theorem) Study-unit Aims The aim of this study-unit is to give a connected and simple account of the most essential parts of the classical dimension theory of separable metric spaces. Students will be exposed to several classical results such as Sperner's lemma and Brouwer Fixed-Point theorem which have many applications also outside dimension theory and topology. The study-unit will develop a topological invariant of Euclidean spaces --- the so called Lebesgue's dimension, with the help of which it will be shown that the Euclidean n-space and the Euclidean m-space are not homeomorphic unless n equals m. Another notable result the students will see is the Nobeling-Pontryagin theorem that every n-dimensional separable metric space is a subset of the Euclidean (2n+1)-space. Finally, it will be also given a brief account of other dimension functions which are all equal to Lebesgue's dimension in the realm of separable metric spaces, but already in the framework of arbitrary metric spaces they are different. Learning Outcomes 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: (a) Understand basic combinatorial techniques and such fundamental concepts as simplex, simplicial complex, triangulation and geometric realization; (b) Know more about geometric properties of Euclidean spaces, and their special subsets (such as spheres, balls, cubes, simplexes); (c) See applications of function spaces and such fundamental topological concepts as compactness, completeness and Baire property. 2. Skills: By the end of the study-unit the student will be able to: (a) Have a rough idea of the proofs of several classical results (such as Sperner's lemma, Brouwer Fixed-Point theorem, Non-Retraction theorem, Non-Homogeneity theorem, Knaster–Kuratowski–Mazurkiewicz theorem, Nobeling-Pontryagin theorem) and the methods used in these proofs; (b) Improve the handling of several topological techniques by seeing how abstract concepts are deployed in particular situations; (c) Have a foundation to follow related more advanced courses. Main Text/s and any supplementary readings Lecture notes covering all topics. Textbooks: • Jan van Mill, Infinite-Dimensional Topology, Prerequisites and Introduction, Volume 43 (North-Holland Mathematical Library), Elsevier Science Publisher B.V., 1989 • Jan van Mill, The Infinite-Dimensional Topology of Function Spaces, Volume 64 (North-Holland Mathematical Library), Elsevier Science Publisher B.V., 2001 |
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| ADDITIONAL NOTES | Pre-Requisite Study-unit: MAT3215 - Metric Spaces. | ||||||
| STUDY-UNIT TYPE | Lecture and Independent Study | ||||||
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| LECTURER/S | Valentin Gutev |
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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 2024/5. It may be subject to change in subsequent years. |
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