Compactifications of Metric Spaces

Compactifications in mathematics date back to the late 19th and early 20th centuries. Maurice Fr´echet and Felix Hausdorff both laid the foundational work on compact spaces, the concept that every open cover has a finite subcover. The notion of compactification emerged as an extension of these compact spaces. The motivation behind compactifications was to find ways to extend a given space in a topological sense by adding limit points or ’points at infinity’. The process of compactification in topology entails changing a given topological space into a compact space, and there exist various methods to achieve this goal. The idea underlying compactifications is the embedding of the original topological space into a compact one. Within the scope of this thesis, our focus lies specifically on metric compactifications, which involve the embedding of metric spaces into compact spaces. Notably, in the case of the real numbers, this process entails the addition of the points ? and ??. In the...Show moreCompactifications in mathematics date back to the late 19th and early 20th centuries. Maurice Fr´echet and Felix Hausdorff both laid the foundational work on compact spaces, the concept that every open cover has a finite subcover. The notion of compactification emerged as an extension of these compact spaces. The motivation behind compactifications was to find ways to extend a given space in a topological sense by adding limit points or ’points at infinity’. The process of compactification in topology entails changing a given topological space into a compact space, and there exist various methods to achieve this goal. The idea underlying compactifications is the embedding of the original topological space into a compact one. Within the scope of this thesis, our focus lies specifically on metric compactifications, which involve the embedding of metric spaces into compact spaces. Notably, in the case of the real numbers, this process entails the addition of the points ? and ??. In the first chapter, we introduce some fundamental concepts and results that are necessary for this thesis. Then, we define the compactification of metric spaces in the second chapter, where we will give an example of the real numbers. We will also demonstrate how we can extend isometries to homeomorphisms on metric compactifications. In the final chapter, we consider the metric compactification of the Euclidean d-dimensional space equipped with the p-norm. This compactification is particularly interesting due to the fact that we can explicitly compute the ’points at infinity’. To determine these points, we utilize the fact that the space is metrizable.Show less

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Compactifications of Metric Spaces