| CODE | MAT3212 | ||||||||
| TITLE | Further Metric Spaces - Completion and Dimension | ||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 6 | ||||||||
| ECTS CREDITS | 5 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | The study-unit is divided into two main sections. The first section gives two methods to construct the 鈥渃ompletion鈥� of a metric space and studies countable products of metric spaces. The second section gives the most essential parts of the classical dimension theory of separable metric spaces. 1) Countable products of Metric Spaces a) The Hilbert Cube I^\omega 2) Completion of a Metric Space a) Uniform convergence and spaces of maps b) Construction of completion using C*(M) c) Construction of completion using Cauchy sequences 3) Fundamental Results on Classical Dimension Theory for Separable Metric Spaces a) The small inductive dimension ind and zero dimensional metric spaces b) The large inductive dimension Ind and the covering dimension dim; their equality to ind in separable metric spaces c) The dimensional properties of Euclidean spaces. The fundamental result that ind R^n = Ind R^n = dim R^n = n. Study-unit Aims: The aim of this study-unit is to further study the concept of a Metric Space, introduced earlier in the course. It is intended to give a construction of the completion of a metric space and see this done in two ways. The notion of a product of metric spaces is now defined for a countable number of metric spaces, compared to that of only a finite number of spaces. The study-unit aims to provide the most essential parts of the classical dimension theory of separable metric spaces. The dimension of a metric space X is defined in three different ways; the small inductive dimension ind(X), the large inductive dimension Ind(X), and the covering dimension dim(X). The three dimension functions coincide in the class of separable metric spaces, i.e., ind(X) = Ind(X) = dim(X) for every separable metric space X Learning Outcomes: 1. Knowledge & Understanding By the end of the study-unit the student will be able to: (a) Work with the definition of the small inductive dimension ind; by having seen simple consequences and reformulations of the definition; (b) Compare the properties of zero-dimensional spaces with the properties of different highly disconnected spaces; (c) Compare the large inductive dimension Ind and the covering dimension dim and see that they both coincide with the small inductive dimension ind in the class of separable metric spaces; (d) Describe the fundamental theorem of dimension theory, which states that ind(R^n) = Ind(R^n) = dim(R^n) = n; (e) Work with countable products of Metric Spaces and see the construction of the completion of a metric space. 2. Skills By the end of the study-unit the student will be able to: a) Construct proofs of several classical results in the dimension theory of separable metric spaces illustrating the mathematical concepts presented in the study-unit; (b) Explain the reasoning of these proofs through rigorous analysis clearly and precisely, using appropriate technical language; (c) Produce important examples of metric spaces, such as spaces of maps and countable products of known metric spaces; (d) Be prepared to follow related more advanced courses. Main Text/s and any supplementary readings: Lecture notes covering all topics in detail. Textbooks (Not Available): 鈥� Ryszard Engelking, Dimension Theory, Volume 19 (North-Holland Mathematical Library), North-Holland Publishing Company, Amsterdam and New York; Polish Scientific Publishers, Warsaw, 1978. 鈥� Ryszard Engelking and Karol Sieklucki, Topology - A Geometric Approach, Volume 4 (Sigma Series in Pure Mathematics), Heldermann Verlag Berlin, 1992. |
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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 | David Buhagiar |
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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 2025/6. It may be subject to change in subsequent years. |
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