Geodesic
The minimum uniform compactification of a metric space
Fundamenta Mathematicae, Tome 147 (1995) no. 1, pp. 39-59
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It is shown that associated with each metric space (X,d) there is a compactification $u_dX$ of X that can be characterized as the smallest compactification of X to which each bounded uniformly continuous real-valued continuous function with domain X can be extended. Other characterizations of $u_dX$ are presented, and a detailed study of the structure of $u_dX$ is undertaken. This culminates in a topological characterization of the outgrowth $u_dℝ^n ∖ ℝ^n$, where $(ℝ^n,d)$ is Euclidean n-space with its usual metric. 
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R. Grant Woods. The minimum uniform compactification of a metric space. Fundamenta Mathematicae, Tome 147 (1995) no. 1, pp. 39-59. doi: 10.4064/fm-147-1-39-59

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