MTH6127 - Metric Spaces and Topology - 2023/24
Topic outline
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This exam paper is of a similar style and standard as this year's exam.
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We have learnt how to measure distance on the real line (using the absolute value) and between two points on the plane (applying the Pythagorean Theorem). But can we measure distance between two vectors in a multidimensional Euclidean space, or between two square matrices, or perhaps two functions? The answer is, yes, we can.
In this module we study metric spaces which are sets of mathematical objects, such as numbers, vectors, matrices, and functions, equipped with the geometric concept of distance (metric). Inside the universe of a Metric Space, we shall generalize the concepts of convergence and continuity, ideas studied in real analysis and explore the foundations of continuous mathematics. We shall discuss Fixed Point Theorems which play an important role for proving the existence of solutions of differential equations and equilibrium points in economic markets.
And here is a more intriguing question. Can we define continuity and convergence without relying on the concept of distance? The answer is again yes, with the introduction of a topology (τόπος – λόγος, meaning the study of location). The study of topological spaces, a further abstraction of metric spaces, will be our final destination in this module. -
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- Definition of metric space; examples, including finite metric spaces, function spaces, normed vector spaces, inner product spaces.
- Topology of metric spaces; open and closed sets, properties, equivalent metrics.
- Topological spaces; examples, properties, dense subsets, Hausdorff spaces.
- Convergence; limit points, closed sets.
- Completeness; Cauchy sequences, properties, Cantor's theorem, examples, Banach and Hilbert space, completion, Lebesgue spaces.
- Continuity; examples, sequential continuity, distance to a set, uniform continuity, homeomorphisms.
- Connectedness; properties, examples.
- (Sequential) compactness, properties of compact spaces, uniform continuity, Heine-Borel Theorem, Bolzano-Weierstrass Theorem.
- Contraction mappings, Banach fixed-point theorem.
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This module covers:
- Definition and examples of metric spaces and topological spaces and basic concepts associated with them.
- Convergence, compactness, continuity and connectedness.
- Completeness of metric spaces, contraction mappings and applications.
DISCIPLINARY SKILLS
At the end of this module, students should be able to:
- Define a metric space and topological space and their associated properties.
- Recognise the properties of a metric space and topological space in specific examples.
- Interpret concepts from analysis of a single real variable (convergence, continuity) in the context of metric spaces and topological spaces.
- Define important concepts such as compactness, connectedness and completeness, recognise them in concrete examples, and use them to derive conclusions.
ATTRIBUTES
At the end of this module, students should have developed with respect to the following attributes:
- Acquire and apply knowledge in a rigorous way.
- Acquire an ability to abstract familiar concepts.
- Adapt their understanding to new and unfamiliar settings.
- Explain and argue clearly and concisely.
- Apply their analytical skills to investigate unfamiliar problems.
- Acquire substantial bodies of new knowledge.
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