Metric spaces and continuity
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This free OpenLearn course, Metric spaces and continuity, is an extract from the Open University course, a third level course that introduces important topics in the theory of pure mathematics including: number theory; the algebraic theory of rings and fields; and metric spaces. Students studying M303 develop their understanding of group theory and real analysis and see how some of the ideas are applied to cryptography and fractals.
Metric spaces and continuity consists of material from M303 Book D, Chapter 14 and has three sections in total. The whole extract should take about 16 hours to study. Section 1 is longer than the other two sections and so you should plan to spend a significant proportion of your time studying Section 1. The extract is a small part (around 4%) of a large course that is studied over eight months, and so can give only an approximate indication of the level and content of the full course.
The Euclidean distance function for the plane has three properties: non-negativity, symmetry and the Triangle Inequality. In a similar way, we find in this extract that the Euclidean distance on Rn satisfies the same three properties. The creative leap is to use this observation to raise these properties to the status of axioms and to say that any function that satisfies them is a well-defined distance function, known as a metric. The extract is relatively self-contained and should be reasonably easy to understand for someone with a sound knowledge of pure mathematics, such as could be gained from studying the Open University course M208 Pure mathematics. A few techniques and definitions are present in the extract without explanation.
Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. However, the extract isn't entirely representative of the module materials, because there are no explicit references to use of the M303 video material (although please note that the PDF may contain references to other parts of M303). In this extract, some illustrations have also been removed due to copyright restrictions.
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Section 1 introduces the idea of a metric space and shows how this concept allows us to generalise the notion of continuity.
Section 2 develops the idea of sequences and convergence in metric spaces.
Section 3 builds on the ideas from the first two sections to formulate a definition of continuity for functions between metric spaces.