Geodesic
Basic Theorems of the~Theory of Pre-ends for Space Domains
Matematičeskie trudy, Tome 1 (1998) no. 2, pp. 79-110

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In the article, we present two constructions based on the theory of pre-ends for domains in the Euclidean space $\mathbb R^n$. The former, using factorization of a set, allows us to compactify every domain in $\mathbb R^n$ homeomorphic to a ball. Applying the latter, we obtain the Hausdorff completion of a domain. As a result, we establish some analogs of the Carathéodory theorem on the boundary behavior of quasi-isometries. 
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A. P. Karmazin. Basic Theorems of the~Theory of Pre-ends for Space Domains. Matematičeskie trudy, Tome 1 (1998) no. 2, pp. 79-110. http://geodesic.mathdoc.fr/item/MT_1998_1_2_a3/